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Algebra II

Algebra II Study Guide Expanded Edition

11 Units · 220 Quiz Questions · 47 Flashcards · Diagnostic · Worked Examples · Full-Length Practice Exam

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Dividing polynomials
Polynomial division works like long division with numbers — you can also use a shortcut called synthetic division when dividing by (x − c).
  • Long division: divide the leading terms, multiply back, subtract, and bring down the next term — repeat until the remainder's degree is less than the divisor's.
  • Synthetic division only works when dividing by a linear factor (x − c). It's faster but gives the same quotient and remainder as long division.
  • A division problem can always be checked: (divisor)(quotient) + remainder = original dividend.
The Remainder Theorem
A shortcut for evaluating a polynomial at a specific value without fully dividing.
  • If a polynomial P(x) is divided by (x − c), the remainder equals P(c).
  • This means you can find P(c) either by direct substitution OR by synthetic division — both give the same number.
  • Useful for checking synthetic division work: the remainder you compute should match P(c) from substitution.
The Factor Theorem
A special case of the Remainder Theorem that tells you whether (x − c) is a factor.
  • (x − c) is a factor of P(x) if and only if P(c) = 0.
  • If P(c) = 0, then c is a zero (root) of the polynomial, and (x − c) divides it evenly (remainder 0).
  • Used to test possible rational roots before fully factoring a higher-degree polynomial.
Zeros, multiplicity & end behavior
The zeros of a polynomial and how it behaves as x gets very large or very small (in either direction) are both controlled by its factored form.
  • A zero's multiplicity is how many times its factor repeats. Odd multiplicity → the graph crosses the x-axis. Even multiplicity → the graph touches and bounces off.
  • End behavior is controlled by the degree and the leading coefficient: odd degree → ends point in opposite directions; even degree → both ends point the same direction.
  • A positive leading coefficient means the right end goes up; a negative leading coefficient means the right end goes down.
Synthetic division only works for a LINEAR divisor (x − c) — you cannot use it to divide by a quadratic.
When using synthetic division for (x + 3), remember c = −3, not 3 — the sign flips from how the factor is written.
Even multiplicity means the graph TOUCHES the x-axis and bounces back, it does NOT cross through.
Remainder Theorem: P(x) ÷ (x−c) has remainder P(c)
Factor Theorem: (x−c) is a factor ⟺ P(c) = 0
Check: (divisor)(quotient) + remainder = dividend
Diagram
down up

End behavior of an odd-degree polynomial with a positive leading coefficient: down on the far left, up on the far right.

Simplifying rational expressions
A rational expression is a fraction with polynomials in the numerator and denominator — simplify by factoring first.
  • Factor both the numerator and denominator completely, then cancel any common factors.
  • Excluded values are any x that make the ORIGINAL denominator zero — these must always be stated, even after simplifying.
  • You can only cancel common FACTORS, never individual terms being added or subtracted.
Multiplying & dividing rational expressions
Works exactly like multiplying and dividing regular fractions, just with polynomials.
  • To multiply, factor everything first, then cancel common factors across the numerators and denominators before multiplying straight across.
  • To divide, multiply by the reciprocal of the second fraction (flip it), then proceed as with multiplication.
  • Always factor before canceling — never cancel terms that are still part of a sum or difference.
Adding & subtracting rational expressions
Just like regular fractions, you need a common denominator before combining.
  • Find the least common denominator (LCD) by factoring each denominator completely.
  • Rewrite each fraction with the LCD, combine the numerators, then simplify if possible.
  • The LCD is NOT just the product of the denominators if they share common factors — factor first to avoid unnecessary extra factors.
Solving rational equations
Clear the denominators to turn a rational equation into a polynomial equation you already know how to solve.
  • Multiply every term by the LCD to eliminate all denominators, then solve the resulting equation.
  • ALWAYS check your solution(s) against the excluded values (anything that made an original denominator zero) — an excluded value that appears as a 'solution' must be rejected.
  • A rejected solution is called extraneous, just like with radical equations.
Forgetting to state excluded values (where the original denominator is zero) — this is a frequently tested step, not optional.
Canceling individual terms instead of common FACTORS — you can only cancel things that are multiplied, never things being added or subtracted.
Forgetting to check for extraneous solutions after clearing denominators in a rational equation.
Excluded values: any x making the ORIGINAL denominator = 0
Multiply by the reciprocal to divide rational expressions
Clear denominators by multiplying every term by the LCD
Simplifying radicals with variables
The same perfect-square (or perfect-cube, etc.) factoring idea from Algebra I extends to expressions with variables.
  • √(x⁶) = x³ — divide the exponent by 2 (the index) when the result is a whole number.
  • For a variable to an odd power, pull out the largest even power first: √(x⁵) = √(x⁴·x) = x²√x.
  • Higher-index roots work the same way: for ∛(x⁹), divide the exponent by 3, giving x³.
Operations with radicals
Adding, subtracting, and multiplying radicals with variables follows the exact same like-radical rules as before.
  • Only radicals with the same index AND the same expression underneath can be combined by addition or subtraction.
  • To multiply radicals with the same index, multiply what's under the roots together, then simplify: ∛4 · ∛2 = ∛8 = 2.
  • Rationalizing a denominator means eliminating a radical from the denominator by multiplying top and bottom by an appropriate radical (or conjugate).
Rational exponent laws
All the integer exponent laws (product, quotient, power rules) apply exactly the same way to rational (fractional) exponents.
  • x^(1/n) means the nth root of x. x^(m/n) means the nth root of x, raised to the m power (or vice versa — same result).
  • To simplify (x^(2/3))^(3/4), multiply the exponents: 2/3 × 3/4 = 1/2, giving x^(1/2) = √x.
  • Negative rational exponents still flip to the reciprocal: x^(−1/2) = 1/x^(1/2) = 1/√x.
Solving radical equations
With higher-index roots or multiple radical terms, the strategy is the same: isolate, raise to a power, and always check.
  • Isolate one radical term, then raise both sides to the power that matches the radical's index (square for a square root, cube for a cube root).
  • If two radical terms remain, you may need to isolate and raise to a power more than once.
  • Cube roots (and other odd-index roots) never introduce extraneous solutions the way square roots can — but checking is still good practice.
√(x⁵) is NOT x^(5/2) rounded — it must be split as x²√x, keeping a radical in the simplified answer.
Rationalizing a denominator with a BINOMIAL radical (like 3 + √2) requires multiplying by the conjugate (3 − √2), not just √2 alone.
When raising both sides of an equation to an EVEN power (like squaring), you must check for extraneous solutions; raising to an ODD power (like cubing) does not create this issue.
x^(m/n) = ⁿ√(xᵐ)
Rationalize a binomial denominator using the conjugate
Even-power operations can introduce extraneous solutions — always check
Exponential functions & the number e
Exponential functions model constant percent growth or decay; e ≈ 2.718 is a special base that shows up naturally in continuous growth.
  • General form: y = a·bˣ, where a is the initial value and b is the growth (b>1) or decay (0
  • Continuous growth/decay uses base e: y = a·e^(rt), where r is the continuous rate.
  • The graph of an exponential function has a horizontal asymptote and never actually touches the x-axis.
Logarithms as inverses
A logarithm answers the question: 'what exponent do I need?' It undoes an exponential exactly the way a square root undoes a square.
  • logₐ(x) = y means aʸ = x. The log tells you the EXPONENT needed on base a to get x.
  • log(x) with no base written means base 10 (common log). ln(x) means base e (natural log).
  • Because logs and exponentials are inverses, log_a(a^x) = x and a^(log_a(x)) = x.
Properties of logarithms
These properties let you expand or combine logarithmic expressions — they mirror the exponent rules exactly.
  • Product Rule: logₐ(MN) = logₐM + logₐN — a log of a product becomes a SUM of logs.
  • Quotient Rule: logₐ(M/N) = logₐM − logₐN — a log of a quotient becomes a DIFFERENCE of logs.
  • Power Rule: logₐ(Mᵖ) = p·logₐM — an exponent inside a log can be pulled out front as a multiplier.
Solving exponential & logarithmic equations
Use logs to solve for a variable trapped in an exponent, and use exponentials to solve for a variable trapped inside a log.
  • To solve for x in an exponential equation like 2ˣ = 20, take the log of both sides: log(2ˣ) = log(20), then use the power rule to bring x down: x·log(2) = log(20), so x = log(20)/log(2).
  • To solve a logarithmic equation like log₂(x) = 5, rewrite in exponential form: x = 2⁵ = 32.
  • Always check log-equation solutions — you cannot take the log of a negative number or zero, so some algebraic solutions must be rejected.
logₐ(M + N) is NOT the same as logₐM + logₐN — the product rule only applies to multiplication inside the log, never addition.
log(x) with no base means base 10; ln(x) means base e — mixing these up leads to wrong calculator button presses.
You cannot take the logarithm of a negative number or zero — always check that your final answer keeps every log argument positive.
logₐ(x) = y ⟺ aʸ = x
Product: logₐ(MN)=logₐM+logₐN · Quotient: logₐ(M/N)=logₐM−logₐN · Power: logₐ(Mᵖ)=p·logₐM
y = a·bˣ (growth: b>1, decay: 0<b<1) · continuous: y = a·e^(rt)
Arithmetic and geometric sequences (review)
A sequence lists terms one at a time; a series is the SUM of a sequence's terms.
  • Arithmetic sequence: add a common difference d each step. aₙ = a₁ + (n−1)d.
  • Geometric sequence: multiply by a common ratio r each step. aₙ = a₁ · r^(n−1).
  • A sequence can also be written recursively — defining each term based on the term(s) before it, plus a starting value.
Sigma (summation) notation
Sigma notation is compact shorthand for writing out and adding a list of terms.
  • Σ (from i=1 to n) of aᵢ means 'add up the terms aᵢ, starting at i=1 and ending at i=n.'
  • The variable below Σ is the index — it changes with each term; the number on top is the last value the index takes.
  • To evaluate, plug in each value of the index one at a time, then add all the results together.
Arithmetic series
There's a fast formula for adding up an arithmetic sequence without writing out every term.
  • Sum of the first n terms: Sₙ = n/2 · (a₁ + aₙ), or equivalently, n times the average of the first and last term.
  • This works because pairing the first and last term, second and second-to-last, etc., always gives the same sum.
  • You need to know (or find) the last term aₙ before applying this formula.
Geometric series
Geometric series have their own sum formula, and — uniquely — an INFINITE geometric series can have a finite sum.
  • Sum of the first n terms: Sₙ = a₁(1 − rⁿ)/(1 − r), for r ≠ 1.
  • An infinite geometric series converges (has a finite sum) ONLY when |r| < 1: S = a₁/(1 − r).
  • If |r| ≥ 1, the infinite series has no finite sum — it diverges.
An infinite geometric series only has a sum when |r| < 1 — if |r| ≥ 1, there is NO sum (it diverges), a very common trap.
Mixing up the arithmetic series formula (uses a₁ and aₙ, the average) with the geometric series formula (uses r, a ratio) — check which type of sequence you have first.
In sigma notation, always double check the STARTING value of the index — it isn't always 1.
Arithmetic series: Sₙ = n/2 · (a₁ + aₙ)
Geometric series (finite): Sₙ = a₁(1−rⁿ)/(1−r)
Geometric series (infinite, |r|<1): S = a₁/(1−r)
Radian measure
Radians are an alternate way to measure angles, based on the radius of a circle instead of degrees.
  • One full revolution = 360° = 2π radians. Halfway around = 180° = π radians.
  • To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π.
  • Radians are the 'natural' unit for trig functions in higher math — many formulas assume angles are in radians.
The unit circle
The unit circle (radius 1, centered at the origin) is the foundation for defining trig functions for ANY angle, not just those in a right triangle.
  • For any angle θ measured from the positive x-axis, the terminal point on the unit circle has coordinates (cos θ, sin θ).
  • Key angles to memorize: 0°, 30°, 45°, 60°, 90° (and their radian equivalents 0, π/6, π/4, π/3, π/2) along with their exact coordinates.
  • Angles are measured counterclockwise from the positive x-axis for positive values.
The six trig functions
Beyond sine and cosine, there are four more trig functions built from ratios of x, y, and r on the unit circle.
  • sin θ = y, cos θ = x (on the unit circle, since r = 1). tan θ = y/x = sin θ / cos θ.
  • The reciprocal functions: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
  • tan θ and cot θ are undefined wherever their denominator (cos θ or sin θ) equals 0.
Reference angles
A reference angle lets you find trig values for ANY angle using only the special first-quadrant angles you've memorized.
  • A reference angle is the acute angle between the terminal side and the x-axis — always between 0° and 90°.
  • Find the trig value using the reference angle's magnitude, then apply the correct sign based on which quadrant the original angle is in.
  • Quadrant signs (ASTC, 'All Students Take Calculus'): Quadrant I all positive, II only sine positive, III only tangent positive, IV only cosine positive.
Forgetting to switch your calculator to RADIAN mode when working with radian-measured angles (or DEGREE mode when working in degrees) — this silently gives wrong answers.
The reference angle is always POSITIVE and between 0° and 90°, even for angles in other quadrants.
tan θ is undefined wherever cos θ = 0 (at 90°, 270°, etc.) — a common oversight when analyzing tangent graphs.
Degrees → radians: multiply by π/180 · Radians → degrees: multiply by 180/π
On the unit circle: cos θ = x, sin θ = y, tan θ = y/x
ASTC: quadrant I all +, II sin only, III tan only, IV cos only
Diagram
(1, 0), 0° (0, 1), 90° (−1, 0), 180° (0, −1), 270° 60°

The unit circle with the four quadrant special angles: 0°, 90°, 180°, 270° and their coordinates.

Graphing sine and cosine
The graphs of y = sin x and y = cos x are smooth waves — four numbers control every transformation of that wave.
  • For y = A sin(Bx − C) + D: |A| is the amplitude (height from midline to peak), period = 2π/B (length of one full cycle).
  • C/B is the phase shift (horizontal shift), and D is the vertical shift, which also gives the new midline y = D.
  • sin x starts at the midline going up; cos x starts at its maximum — this is the key visual difference between the two graphs.
The Pythagorean identity
The most important trig identity, coming directly from the equation of the unit circle (x² + y² = 1).
  • sin²θ + cos²θ = 1, true for every angle θ.
  • Two related identities can be derived by dividing through by cos²θ or sin²θ: 1 + tan²θ = sec²θ, and 1 + cot²θ = csc²θ.
  • These identities let you find one trig value if you know another, without needing the actual angle.
Reciprocal & quotient identities
These identities directly restate how the six trig functions relate to each other.
  • Reciprocal identities: csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ.
  • Quotient identities: tan θ = sin θ / cos θ, and cot θ = cos θ / sin θ.
  • These are used constantly to rewrite an expression entirely in terms of sine and cosine, which often makes simplifying easier.
Solving trigonometric equations
Solving a trig equation means finding every angle that makes it true — usually within one full rotation (0 to 2π or 0° to 360°), unless told otherwise.
  • Isolate the trig function first, just like isolating a variable, then use the unit circle (or inverse trig) to find the reference angle.
  • Use the reference angle and the quadrant signs (ASTC) to find ALL solutions in the given interval — most trig equations have more than one solution.
  • If no interval is given, add '+ 2πk' (or '+ 360°k') to represent every possible coterminal solution, where k is any integer.
Forgetting that a trig equation on [0, 2π) usually has MULTIPLE solutions — not just the one your calculator's inverse function gives you.
sin²θ + cos²θ = 1 is an IDENTITY (true for all θ), not an equation to 'solve' — don't try to isolate θ from it directly.
Mixing up amplitude (|A|, the height) with period (2π/B, the horizontal length of one cycle) when reading a graph's equation.
sin²θ + cos²θ = 1 · 1+tan²θ=sec²θ · 1+cot²θ=csc²θ
y = A sin(Bx−C)+D: amplitude |A|, period 2π/B
tan θ = sin θ/cos θ · cot θ = cos θ/sin θ
Diagram
amplitude = 1 period = 2π

y = sin(x): amplitude 1, period 2π. The curve starts at the midline and reaches its max a quarter of the way through one period.

The imaginary unit i
i is defined so that negative numbers can have square roots — it's the building block of every complex number.
  • i = √(−1), so i² = −1.
  • √(−n) = i√n for any positive number n. Example: √(−9) = 3i.
  • Powers of i cycle in a pattern of 4: i¹=i, i²=−1, i³=−i, i⁴=1, then it repeats.
Operations with complex numbers
A complex number has the form a + bi, where a is the real part and b is the imaginary part — add, subtract, and multiply them like binomials, using i²=−1.
  • Add/subtract by combining the real parts together and the imaginary parts together.
  • Multiply using FOIL, then simplify using i² = −1.
  • The complex conjugate of a + bi is a − bi. Multiplying a complex number by its conjugate always gives a real number: (a+bi)(a−bi) = a² + b².
Complex solutions to quadratics
When a quadratic's discriminant is negative, its solutions are complex numbers instead of real ones — the quadratic formula still works exactly the same way.
  • A negative discriminant (b² − 4ac < 0) means the parabola never crosses the x-axis — its two solutions are a complex conjugate pair.
  • Apply the quadratic formula as usual; when you reach a negative number under the square root, rewrite it using i.
  • Complex solutions to a quadratic with real coefficients always come in conjugate pairs: if a+bi is a solution, so is a−bi.
Dividing complex numbers
To divide by a complex number, eliminate the imaginary part of the denominator using the conjugate — very similar to rationalizing a radical denominator.
  • Multiply the numerator and denominator by the conjugate of the denominator.
  • The denominator becomes a real number (using (a+bi)(a−bi) = a²+b²), leaving a simplified complex number.
  • Distribute in the numerator carefully, then simplify using i² = −1.
i² = −1, not +1 — this single substitution is the key to every complex-number simplification.
The powers of i cycle every 4 steps — to simplify a high power like i²³, divide the exponent by 4 and use the remainder (23 ÷ 4 = 5 remainder 3, so i²³ = i³ = −i).
Complex solutions to a real-coefficient quadratic ALWAYS come in conjugate pairs — a single, unpaired complex solution signals an arithmetic error.
i² = −1 · √(−n) = i√n
Conjugate of a+bi is a−bi · (a+bi)(a−bi) = a²+b²
Powers of i repeat every 4: i,−1,−i,1,...
Diagram
Real Imaginary 3 + 4i

The complex number 3 + 4i plotted on the complex plane: 3 units along the real axis, 4 units along the imaginary axis.

Operations on functions
Two functions can be combined with the same four operations used on numbers, plus a fifth special operation: composition.
  • (f+g)(x) = f(x)+g(x), (f−g)(x) = f(x)−g(x), (f·g)(x) = f(x)·g(x), (f/g)(x) = f(x)/g(x) (with g(x)≠0).
  • Composition (f∘g)(x) = f(g(x)) means: first evaluate g at x, then plug THAT result into f.
  • Composition is generally NOT commutative: (f∘g)(x) usually does not equal (g∘f)(x).
Inverse functions
An inverse function 'undoes' the original function — swapping every input and output.
  • To find an inverse algebraically: swap x and y in the equation, then solve for the new y.
  • f(x) and f⁻¹(x) satisfy f(f⁻¹(x)) = x and f⁻¹(f(x)) = x for all valid x.
  • A function only has an inverse that is ALSO a function if the original passes the Horizontal Line Test (each y-value comes from only one x-value).
Verifying inverses graphically
There's a fast visual check for whether two functions are inverses of each other.
  • The graphs of f(x) and f⁻¹(x) are always reflections of each other across the line y = x.
  • If a point (a, b) is on the graph of f, then the point (b, a) must be on the graph of f⁻¹.
  • This reflection relationship is why swapping x and y algebraically produces the inverse.
Transformations, combined
Multiple transformations can be applied to a function at once — order and grouping matter.
  • g(x) = A·f(B(x − h)) + k combines a vertical stretch/reflection (A), horizontal stretch/reflection (B), horizontal shift (h), and vertical shift (k) all at once.
  • Apply transformations in this order (matching order of operations): horizontal shift and stretch happen INSIDE the function first, then the outside stretch and vertical shift are applied to the result.
  • A negative A reflects over the x-axis (flips vertically); a negative B reflects over the y-axis (flips horizontally).
(f∘g)(x) means g happens FIRST, then f — reading it left to right (f first) is backwards.
Not every function has an inverse function — the original must pass the Horizontal Line Test, or its inverse relation won't be a function.
When combining transformations, horizontal changes (inside the parentheses) behave opposite to how they look: g(x−h) shifts RIGHT, not left.
(f∘g)(x) = f(g(x)) — apply g first, then f
Inverse check: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x
Graphs of f and f⁻¹ reflect across y = x
Diagram
y = x f(x) f⁻¹(x)

A function f and its inverse f⁻¹ are always mirror images of each other across the line y = x.

Sampling methods
How a sample is collected determines whether it's safe to generalize its results to a larger population.
  • Simple random sample: every member of the population has an equal chance of being selected — the gold standard for avoiding bias.
  • Stratified sample: the population is divided into subgroups (strata), then a random sample is taken from each subgroup proportionally.
  • A biased sampling method (like only surveying volunteers, or only one location) can make results unreliable, no matter how large the sample is.
The normal distribution
Many real-world data sets follow a symmetric, bell-shaped curve called the normal distribution, centered at the mean.
  • The Empirical Rule (68-95-99.7 Rule): about 68% of data falls within 1 standard deviation of the mean, 95% within 2, and 99.7% within 3.
  • A z-score tells you how many standard deviations a value is from the mean: z = (x − μ) / σ.
  • A positive z-score means the value is above the mean; a negative z-score means it's below.
Margin of error & confidence intervals
Since a sample can never perfectly represent an entire population, statisticians report a range of plausible values instead of one exact number.
  • Margin of error accounts for natural sampling variability — a bigger sample generally produces a SMALLER margin of error.
  • A confidence interval is: sample statistic ± margin of error, giving a range likely to contain the true population value.
  • A '95% confidence interval' means: if you repeated the sampling process many times, about 95% of the resulting intervals would contain the true population parameter.
Simulation & hypothesis testing basics
Simulations help decide whether an observed result is likely due to random chance or reflects a real difference.
  • A simulation repeats a random process many times to build a distribution of what 'random chance alone' would typically produce.
  • If an observed result falls far outside the range the simulation typically produces, that's evidence the result is NOT just due to random chance.
  • This reasoning is the foundation of statistical significance — deciding whether a result is surprising enough to matter.
A larger sample size reduces margin of error, but it does NOT fix a biased sampling method — bias is a different problem than sample size.
'95% confidence' describes the reliability of the METHOD over many repeated samples, not the probability that one specific interval is correct.
Mixing up which direction a z-score points: positive is ABOVE the mean, negative is BELOW the mean.
z-score: z = (x − μ) / σ
Empirical Rule: 68% within 1σ, 95% within 2σ, 99.7% within 3σ
Confidence interval = sample statistic ± margin of error
Diagram
μ −1σ+1σ 68%

The normal distribution and the Empirical Rule: about 68% of data falls within 1 standard deviation of the mean, 95% within 2, 99.7% within 3.

Basic probability rules
Probability measures how likely an event is, always as a number between 0 (impossible) and 1 (certain).
  • P(event) = (number of favorable outcomes) / (total number of outcomes), when all outcomes are equally likely.
  • P(A or B) = P(A) + P(B) − P(A and B) — this avoids double-counting any overlap between A and B.
  • For mutually exclusive events (they cannot both happen), P(A and B) = 0, so P(A or B) simplifies to just P(A) + P(B).
Independent, dependent & conditional probability
Whether one event affects the probability of another changes how you calculate combined probabilities.
  • Independent events: the outcome of one does NOT affect the other. P(A and B) = P(A) · P(B).
  • Dependent events: the outcome of one DOES affect the other's probability, so you must account for that change.
  • Conditional probability P(B | A) means 'the probability of B, GIVEN that A already happened' — the sample space shrinks to only outcomes where A occurred.
Permutations & combinations
Both count arrangements of items, but permutations care about ORDER while combinations do not.
  • Permutation (order matters): nPr = n! / (n−r)!. Example: arranging 3 of 5 people in a line.
  • Combination (order doesn't matter): nCr = n! / (r!(n−r)!). Example: choosing 3 of 5 people for a committee.
  • A quick test: if rearranging the same items counts as a DIFFERENT outcome, use a permutation; if it counts as the SAME outcome, use a combination.
Binomial probability
Used when a fixed number of independent trials each have only two possible outcomes (success or failure).
  • Requirements: a fixed number of trials n, two outcomes per trial, constant probability of success p, and independent trials.
  • P(exactly k successes) = nCk · pᵏ · (1−p)^(n−k).
  • Example: the probability of getting exactly 3 heads in 5 coin flips uses n=5, k=3, p=0.5.
Forgetting to subtract the overlap P(A and B) when finding P(A or B) for events that CAN happen together — this double-counts the overlap.
Confusing permutations (order matters, more arrangements) with combinations (order doesn't matter, fewer arrangements) — always ask if rearranging counts as 'different.'
P(A and B) = P(A)·P(B) ONLY works for INDEPENDENT events — for dependent events you must use the conditional probability instead.
P(A or B) = P(A) + P(B) − P(A and B)
Independent: P(A and B) = P(A)·P(B)
nPr = n!/(n−r)! · nCr = n!/(r!(n−r)!)
📝 Practice Question Bank — 220 questions
Unit 1: Polynomial Functions & Operations (20)

  1. Divide (x² + 5x + 6) by (x + 2) using synthetic division

    • x + 3, remainder 0
    • x + 3, remainder 6
    • x − 3, remainder 0
    • x + 2, remainder 3

    x²+5x+6 = (x+2)(x+3), so the quotient is x+3 with remainder 0.

  2. If P(x) = x³ − 2x² + x − 5, find P(2) using the Remainder Theorem

    • −3
    • 3
    • −5
    • 5

    P(2) = 8 − 8 + 2 − 5 = −3.

  3. Is (x − 1) a factor of x³ − 6x² + 11x − 6?

    • Yes, P(1) = 0
    • No, P(1) ≠ 0
    • Yes, but only for even-degree polynomials
    • No, factors must be quadratic

    P(1) = 1 − 6 + 11 − 6 = 0, so by the Factor Theorem, (x−1) is a factor.

  4. Divide x³ − 6x² + 11x − 6 by (x − 1) using synthetic division

    • x² − 5x + 6, remainder 0
    • x² + 5x + 6
    • x² − 5x − 6
    • x² − 6x + 5

    Synthetic division with c=1 on 1,−6,11,−6 gives 1,−5,6, remainder 0.

  5. A polynomial has a zero at x = 3 with multiplicity 2. What happens at x = 3 on the graph?

    • Crosses the x-axis
    • Touches and bounces off the x-axis
    • The graph has a hole
    • The graph is undefined

    Even multiplicity means the graph touches the x-axis and bounces back without crossing.

  6. A polynomial has a zero at x = −2 with multiplicity 3. What happens at x = −2?

    • Crosses the x-axis
    • Touches and bounces off the x-axis
    • No effect on the graph
    • Vertical asymptote

    Odd multiplicity means the graph crosses through the x-axis.

  7. What is the end behavior of y = −x⁴ + 3x² − 1?

    • Up on both ends
    • Down on both ends
    • Down left, up right
    • Up left, down right

    Even degree with a negative leading coefficient: both ends point down.

  8. What is the end behavior of y = 2x⁵ − x³ + 4?

    • Down on both ends
    • Up on both ends
    • Down left, up right
    • Up left, down right

    Odd degree with a positive leading coefficient: down on the left, up on the right.

  9. Which value of c would you use in synthetic division to divide by (x + 5)?

    • 5
    • −5
    • 1/5
    • −1/5

    x + 5 = x − (−5), so c = −5.

  10. (x + 3)(x² − x − 2) equals which of the following?

    • x³ + 2x² − 5x − 6
    • x³ − 2x² + 5x + 6
    • x³ + 2x² + 5x − 6
    • x³ − x² − 2x

    Distributing gives x³−x²−2x+3x²−3x−6 = x³+2x²−5x−6.

  11. A degree-4 polynomial can have at most how many real zeros?

    • 2
    • 3
    • 4
    • 5

    A degree-n polynomial has at most n real zeros.

  12. For P(x) = x² − 9, what are the zeros?

    • x = 3 or x = −3
    • x = 9 or x = −9
    • x = 3 only
    • x = 81

    x² − 9 = (x−3)(x+3) = 0, so x = 3 or x = −3.

  13. Using the Factor Theorem, is (x + 2) a factor of x³ + 3x² − 4?

    • Yes, P(−2) = 0
    • No, P(−2) ≠ 0
    • Yes, P(2) = 0
    • No, must check P(2) instead

    P(−2) = −8 + 12 − 4 = 0, so (x+2) is a factor.

  14. Divide (2x² + 7x + 3) by (x + 3) using synthetic division

    • 2x + 1, remainder 0
    • 2x + 1, remainder 3
    • 2x − 1, remainder 6
    • x + 1, remainder 0

    Synthetic division with c=−3 on 2,7,3 gives 2,1, remainder 0.

  15. If P(x) = 2x³ + x² − 5x + 1, find P(−1)

    • 5
    • −5
    • 3
    • −3

    2(−1)³ + (−1)² − 5(−1) + 1 = −2 + 1 + 5 + 1 = 5.

  16. A degree-3 polynomial with a positive leading coefficient has end behavior:

    • Down left, up right
    • Up left, down right
    • Up on both ends
    • Down on both ends

    Odd degree, positive leading coefficient: down on the left, up on the right.

  17. Which of these is a zero of x³ − x² − 4x + 4, based on the Factor Theorem test P(1)?

    • x = 1
    • x = −1
    • x = 4
    • x = −4

    P(1) = 1 − 1 − 4 + 4 = 0, so x=1 is a zero.

  18. A polynomial has zeros at x = 2 (multiplicity 1) and x = −1 (multiplicity 2). How many times does the graph cross the x-axis?

    • Once (at x=2 only)
    • Twice
    • Three times
    • Never

    Odd multiplicity at x=2 crosses; even multiplicity at x=−1 only touches and bounces, not a true crossing.

  19. What is the degree of the polynomial that results from multiplying a degree-2 polynomial by a degree-3 polynomial?

    • 5
    • 6
    • 1
    • Cannot be determined

    Multiplying polynomials adds their degrees: 2+3=5.

  20. Using synthetic division to divide x³+1 by (x+1), what is the quotient?

    • x² − x + 1
    • x² + x + 1
    • x² − x − 1
    • x² + 1

    Synthetic division with c=−1 on 1,0,0,1 gives 1,−1,1, remainder 0: x²−x+1.

Unit 2: Rational Expressions & Equations (20)

  1. Simplify (x² − 9)/(x + 3)

    • x − 3, x ≠ −3
    • x − 3
    • x + 3, x ≠ 3
    • x − 3, x ≠ 3

    (x−3)(x+3)/(x+3) = x−3, with x ≠ −3 stated since that made the original denominator zero.

  2. State the excluded value(s) for (x + 1)/(x² − 4)

    • x = 2 and x = −2
    • x = 4 and x = −4
    • x = −1
    • x = 2 only

    x² − 4 = 0 when x = 2 or x = −2.

  3. Simplify (x² + 5x + 6)/(x² + 4x + 3)

    • (x + 2)/(x + 1)
    • (x + 3)/(x + 1)
    • (x + 2)/(x + 3)
    • (x + 1)/(x + 2)

    Factor: (x+2)(x+3)/((x+1)(x+3)) — cancel (x+3) to get (x+2)/(x+1).

  4. Multiply: (x/(x+2)) · ((x+2)/(3x))

    • 1/3
    • x/3
    • 3
    • x²/(3(x+2))

    Both x and (x+2) cancel, leaving 1/3.

  5. Divide: (2/(x−1)) ÷ (4/(x−1))

    • 1/2
    • 2
    • 8
    • (x−1)/2

    Multiply by the reciprocal: (2/(x−1))·((x−1)/4) = 2/4 = 1/2.

  6. Add: 3/x + 2/x

    • 5/x
    • 5/(2x)
    • 6/x
    • 5/x²

    Same denominator, add numerators: 5/x.

  7. Add: 1/x + 1/(x+1)

    • (2x+1)/(x(x+1))
    • 2/(2x+1)
    • 1/(x(x+1))
    • (x+1)/x²

    LCD is x(x+1): (x+1)/(x(x+1)) + x/(x(x+1)) = (2x+1)/(x(x+1)).

  8. Subtract: 5/(x−2) − 3/(x−2)

    • 2/(x−2)
    • 2
    • 8/(x−2)
    • 2/x

    Same denominator, subtract numerators: 2/(x−2).

  9. Solve for x: 3/x = 6/(x+2)

    • 2
    • −2
    • 0
    • 4

    Cross multiply: 3(x+2)=6x → 3x+6=6x → x=2. Check: x≠0,−2, so x=2 works.

  10. Solve: x/(x−3) = 2 + 3/(x−3)

    • No solution (x = 3 is excluded)
    • x = 3
    • x = 0
    • x = −3

    Multiplying out gives x = 2(x−3)+3 → x=2x−3 → x=3, but x=3 makes the original denominator zero — no solution.

  11. What must you always check after solving a rational equation?

    • That the solution doesn't make any original denominator zero
    • That the solution isn't negative
    • That the solution is a whole number
    • That the solution is less than 10

    Solutions that create a zero denominator in the original equation must be rejected as extraneous.

  12. Simplify: (x²−1)/(x−1) · 1/(x+1)

    • 1
    • x + 1
    • x − 1
    • (x+1)²

    (x²−1)/(x−1) simplifies to (x+1); then (x+1)·1/(x+1) = 1.

  13. Find the LCD of 1/(x²−4) and 1/(x+2)

    • (x−2)(x+2)
    • x²−4 · (x+2)
    • (x+2)²
    • x−2

    x²−4 factors as (x−2)(x+2), which already contains (x+2), so that's the LCD.

  14. Simplify (2x² − 8)/(x² − 4)

    • 2
    • 2/(x+2)
    • 2(x−2)
    • 1/2

    Factor: 2(x²−4)/(x²−4) = 2 (the x²−4 factors cancel entirely).

  15. State the excluded value(s) for 5/(x² − 25)

    • x = 5 and x = −5
    • x = 25 and x = −25
    • x = 5 only
    • x = 0

    x²−25=0 when x=5 or x=−5.

  16. Multiply: ((x+1)/(x−2)) · ((x−2)/(x+3))

    • (x+1)/(x+3)
    • (x+1)(x−2)/(x+3)
    • 1/(x+3)
    • (x−2)/(x+3)

    The (x−2) factors cancel, leaving (x+1)/(x+3).

  17. Divide: (x/4) ÷ (x²/8)

    • 2/x
    • x/2
    • 2x
    • x²/32

    (x/4)·(8/x²) = 8x/(4x²) = 2/x.

  18. Add: 2/(x+1) + 3/(x−1)

    • (5x+1)/((x+1)(x−1))
    • 5/(x²−1)
    • (5x−1)/(x²−1)
    • 5/(2x)

    LCD (x+1)(x−1): 2(x−1)+3(x+1) = 2x−2+3x+3 = 5x+1, over (x+1)(x−1).

  19. Solve: 2/(x+3) = 1/(x−1)

    • x = 5
    • x = −5
    • x = 1
    • x = 3

    Cross multiply: 2(x−1)=1(x+3) → 2x−2=x+3 → x=5.

  20. Solve: (x+2)/(x−4) = 6/(x−4)

    • No solution (x=4 is required but excluded)
    • x = 4
    • x = 6
    • x = −2

    Multiplying by (x−4): x+2=6 → x=4, but x=4 makes the original denominator zero — no solution.

Unit 3: Radicals & Rational Exponents (20)

  1. Simplify √(x⁶)

    • x³
    • x¹²
    • x²
    • 3x

    Divide the exponent by the index: 6÷2=3.

  2. Simplify √(x⁵)

    • x²√x
    • x²·⁵
    • x√x⁴
    • 5x

    x⁵ = x⁴·x, and √(x⁴) = x², leaving x²√x.

  3. Simplify ∛(x⁹)

    • x³
    • x⁶
    • 3x
    • 9x

    Divide the exponent by the index: 9÷3=3.

  4. Simplify: 5√7 + 3√7

    • 8√7
    • 8√14
    • 15√7
    • 8

    Like radicals combine by adding coefficients: (5+3)√7 = 8√7.

  5. Multiply: ∛4 · ∛2

    • 2
    • ∛6
    • 8
    • 6

    ∛4 · ∛2 = ∛8 = 2.

  6. Simplify (x^(2/3))^(3/4)

    • √x
    • x^(3/2)
    • x^(1/6)
    • x²

    Multiply the exponents: (2/3)(3/4) = 1/2, giving x^(1/2) = √x.

  7. Simplify x^(−1/2)

    • 1/√x
    • −√x
    • √x
    • −½x

    A negative exponent flips to the reciprocal: x^(−1/2) = 1/x^(1/2) = 1/√x.

  8. Rationalize the denominator: 1/√5

    • √5/5
    • 1/5
    • 5/√5
    • √5

    Multiply top and bottom by √5: √5/5.

  9. Rationalize the denominator: 2/(3+√2)

    • (6−2√2)/7
    • (6+2√2)/7
    • 2/(9−2)
    • (3−√2)/7

    Multiply by the conjugate (3−√2): 2(3−√2)/(9−2) = (6−2√2)/7.

  10. Solve √(x+5) = x−1

    • x = 4 only
    • x = 4 or x = −1
    • x = −1 only
    • No solution

    Squaring gives x²−3x−4=0 → (x−4)(x+1)=0 → x=4 or x=−1. Checking x=−1 in the original equation fails (extraneous), so x=4 only.

  11. Simplify: x^(1/3) · x^(1/6)

    • x^(1/2)
    • x^(1/18)
    • x^(2/3)
    • x^(1/9)

    Add the exponents: 1/3+1/6 = 2/6+1/6 = 3/6 = 1/2.

  12. Which expression is equivalent to ⁴√(x⁸)?

    • x²
    • x⁴
    • x⁸
    • x^(1/2)

    Divide the exponent by the index: 8÷4=2.

  13. Simplify: √50 − √8

    • 3√2
    • 3√42
    • 42
    • 7√2

    √50=5√2 and √8=2√2, so 5√2−2√2=3√2.

  14. Simplify √(x⁴ · y²)

    • x²|y|
    • x²y
    • xy
    • x⁴y²

    √(x⁴)=x² and √(y²)=|y| (since y could be negative).

  15. Simplify: 2∛16

    • 4∛2
    • 2∛16 (already simplified)
    • 8∛2
    • 4∛4

    16=8·2, so ∛16=2∛2, and 2·2∛2 = 4∛2.

  16. Simplify (x^(3/4))^(4/3)

    • x
    • x^(9/16)
    • x^(7/7)
    • x²

    Multiply the exponents: (3/4)(4/3) = 1, giving x¹ = x.

  17. Solve ∛(x−2) = 3

    • x = 29
    • x = 11
    • x = 25
    • x = 7

    Cube both sides: x−2 = 27, so x = 29.

  18. Simplify: 4√3 · 2√6

    • 24√2
    • 8√18
    • 8√9
    • 24√18

    4·2=8, √3·√6=√18=3√2, so 8·3√2 = 24√2.

  19. Which value of x makes √(x−3) undefined over the real numbers?

    • Any x < 3
    • Any x > 3
    • x = 3
    • x = 0

    The expression under an even-index radical cannot be negative, so x−3 ≥ 0 is required.

  20. Simplify: (9x⁴)^(1/2)

    • 3x²
    • 3x
    • 9x²
    • 3x⁸

    √9=3 and √(x⁴)=x², giving 3x².

Unit 4: Exponential & Logarithmic Functions (20)

  1. Evaluate log₂(8)

    • 3
    • 4
    • 2
    • 8

    2³ = 8, so log₂(8) = 3.

  2. Evaluate log₃(81)

    • 4
    • 3
    • 27
    • 9

    3⁴ = 81, so log₃(81) = 4.

  3. Evaluate log₅(1)

    • 0
    • 1
    • 5
    • Undefined

    5⁰ = 1, so log₅(1) = 0.

  4. Rewrite in exponential form: log₄(x) = 3

    • x = 64
    • x = 12
    • x = 7
    • x = 81

    log₄(x)=3 means 4³=x, so x=64.

  5. Solve log₂(x) = 5

    • x = 32
    • x = 10
    • x = 25
    • x = 7

    2⁵ = 32.

  6. Expand using the product rule: log(xy)

    • log x + log y
    • log x · log y
    • log x − log y
    • log(x+y)

    The product rule turns a log of a product into a sum of logs.

  7. Expand using the quotient rule: log(x/y)

    • log x − log y
    • log x + log y
    • log x / log y
    • log x · log y

    The quotient rule turns a log of a quotient into a difference of logs.

  8. Expand using the power rule: log(x⁵)

    • 5 log x
    • log x⁵ (can't simplify)
    • x log 5
    • log 5 + log x

    The power rule pulls the exponent out front as a multiplier: 5 log x.

  9. Condense: log 3 + log 4

    • log 12
    • log 7
    • log(3/4)
    • log 1

    log 3 + log 4 = log(3·4) = log 12.

  10. Condense: 2 log x − log y

    • log(x²/y)
    • log(2x/y)
    • log(x²−y)
    • log(x²·y)

    Power rule first: log(x²) − log y, then quotient rule: log(x²/y).

  11. Solve for x: 3ˣ = 81

    • 4
    • 27
    • 3
    • 81

    81 = 3⁴, so x = 4.

  12. Solve 2ˣ = 20 to the nearest hundredth

    • 4.32
    • 4.00
    • 10.00
    • 3.32

    x = log(20)/log(2) ≈ 1.301/0.301 ≈ 4.32.

  13. Which base does "ln" represent?

    • e
    • 10
    • 2
    • No fixed base

    ln(x) means logₑ(x), the natural log.

  14. Evaluate log₄(1/16)

    • −2
    • 2
    • −4
    • 1/4

    4⁻²=1/16, so log₄(1/16)=−2.

  15. Solve ln(x) = 2 (round to the nearest hundredth)

    • 7.39
    • 2.72
    • 0.69
    • 5.44

    x = e² ≈ 7.389.

  16. A population grows according to y = 500e^(0.03t). What is the growth rate?

    • 3% continuous growth
    • 30% growth
    • 0.03% growth
    • 500% growth

    In y=ae^(rt), r=0.03 represents a 3% continuous growth rate.

  17. Solve log(x) + log(x−3) = 1

    • x = 5
    • x = −2
    • x = 5 or x = −2
    • x = 3

    log(x(x−3))=1 → x²−3x=10 → x²−3x−10=0 → (x−5)(x+2)=0. x=−2 is rejected (can't take log of a negative), so x=5.

  18. Simplify: log₂(16) + log₂(4)

    • 6
    • 20
    • 10
    • 64

    log₂16=4, log₂4=2, sum=6. (Equivalently log₂(64)=6.)

  19. How long does it take an investment to double at 5% annual compound interest, using y=a(1.05)ᵗ, to the nearest year?

    • 14 years
    • 20 years
    • 10 years
    • 5 years

    Solve 2=1.05ᵗ: t=log(2)/log(1.05)≈14.2, rounds to 14.

  20. Which equation is equivalent to eˣ = 12?

    • x = ln(12)
    • x = log(12)
    • x = 12e
    • x = e/12

    Taking the natural log of both sides: x = ln(12).

Unit 5: Sequences & Series (20)

  1. Find S₁₀ for the arithmetic sequence with a₁ = 4, d = 3

    • 175
    • 155
    • 310
    • 35

    a₁₀ = 4+(9)(3) = 31. S₁₀ = 10/2·(4+31) = 5·35 = 175.

  2. Find the sum of the first 6 terms of the geometric sequence a₁ = 2, r = 3

    • 728
    • 486
    • 364
    • 1458

    S₆ = 2(1−3⁶)/(1−3) = 2(−728)/(−2) = 728.

  3. Find the sum of the infinite geometric series with a₁ = 8, r = 1/2

    • 16
    • 4
    • 8
    • 32

    S = a₁/(1−r) = 8/0.5 = 16.

  4. Does the infinite geometric series with a₁ = 5, r = 2 have a sum?

    • No, because |r| ≥ 1
    • Yes, S = 5
    • Yes, S = 10
    • Yes, S = −5

    An infinite geometric series only converges when |r| < 1. Here |r|=2, so it diverges.

  5. Evaluate Σ (i=1 to 4) of 2i

    • 20
    • 10
    • 24
    • 14

    2(1)+2(2)+2(3)+2(4) = 2+4+6+8 = 20.

  6. Evaluate Σ (i=1 to 3) of (i² + 1)

    • 17
    • 14
    • 20
    • 11

    (1+1)+(4+1)+(9+1) = 2+5+10 = 17.

  7. Find the 8th term of the arithmetic sequence 5, 9, 13, 17, ...

    • 33
    • 37
    • 29
    • 32

    d=4, a₈ = 5+(7)(4) = 33.

  8. Find the 5th term of the geometric sequence 3, 6, 12, 24, ...

    • 48
    • 24
    • 96
    • 32

    r=2, a₅ = 3·2⁴ = 48.

  9. What does Sₙ = n/2·(a₁+aₙ) represent for an arithmetic series?

    • n times the average of the first and last term
    • The product of all terms
    • The common difference times n
    • The nth term alone

    Pairing the first and last term (and so on) always gives the same average, times n terms.

  10. An infinite geometric series has r = −0.5. Does it converge?

    • Yes, since |−0.5| < 1
    • No, since r is negative
    • No, since |r| ≥ 1
    • Only if a₁ > 0

    Convergence depends on |r| < 1, regardless of sign — |−0.5| = 0.5 < 1.

  11. Find S₅ for the geometric sequence a₁ = 1, r = 1/2

    • 31/16
    • 1/32
    • 15/16
    • 2

    S₅ = 1(1−(1/2)⁵)/(1−1/2) = (31/32)/(1/2) = 31/16.

  12. Which recursive formula describes an arithmetic sequence with common difference d?

    • aₙ = aₙ₋₁ + d
    • aₙ = aₙ₋₁ · d
    • aₙ = a₁ + d
    • aₙ = aₙ₋₁ + n

    Each term is the previous term plus the common difference.

  13. For a geometric series with r = 1, which formula gives the sum instead of the standard formula?

    • Sₙ = n·a₁ (every term is equal)
    • Sₙ = a₁(1−1ⁿ)/(1−1)
    • Sₙ = 0
    • Sₙ = a₁ⁿ

    When r=1 the standard formula divides by zero; since every term equals a₁, the sum is just n·a₁.

  14. Find the 20th term of the arithmetic sequence with a₁ = 7, d = −2

    • −31
    • −33
    • 31
    • 45

    a₂₀ = 7+(19)(−2) = 7−38 = −31.

  15. Find the sum of the first 8 terms of the arithmetic sequence 2, 5, 8, 11, ...

    • 100
    • 80
    • 92
    • 110

    d=3, a₈=2+(7)(3)=23. S₈=8/2·(2+23)=4·25=100.

  16. Write the explicit formula for the sequence with a₁ = 10 and r = 1/5

    • aₙ = 10·(1/5)^(n−1)
    • aₙ = 10+(n−1)(1/5)
    • aₙ = 10·5^(n−1)
    • aₙ = (1/5)·10^(n−1)

    Geometric sequence explicit formula: aₙ = a₁·r^(n−1).

  17. Evaluate Σ (i=2 to 5) of 3i

    • 42
    • 45
    • 36
    • 30

    3(2)+3(3)+3(4)+3(5) = 6+9+12+15 = 42.

  18. A ball is dropped and bounces to 60% of its previous height each time, starting at 10 feet. Find the total vertical distance using the infinite geometric series formula for the bounces only (after the drop), a₁=6

    • 15 feet
    • 6 feet
    • 10 feet
    • 16.67 feet

    S = a₁/(1−r) = 6/(1−0.6) = 6/0.4 = 15.

  19. Which sequence is neither arithmetic nor geometric?

    • 1, 4, 9, 16, ...
    • 2, 5, 8, 11, ...
    • 3, 6, 12, 24, ...
    • 10, 7, 4, 1, ...

    1,4,9,16 (perfect squares) has no constant difference or ratio between terms.

  20. Find r for the geometric sequence 100, 20, 4, 0.8, ...

    • 1/5
    • 5
    • 1/4
    • 4

    20/100 = 1/5, and 4/20 = 1/5 — confirmed constant ratio.

Unit 6: Trigonometric Functions & the Unit Circle (20)

  1. Convert 180° to radians

    • π
    • 2π
    • π/2
    • π/4

    180° is half of a full revolution (2π), so it equals π radians.

  2. Convert 90° to radians

    • π/2
    • π
    • π/4
    • 2π

    90° is a quarter revolution: (2π)/4 = π/2.

  3. Convert π/3 radians to degrees

    • 60°
    • 45°
    • 90°
    • 30°

    (π/3)(180/π) = 60°.

  4. Convert 3π/2 radians to degrees

    • 270°
    • 180°
    • 360°
    • 135°

    (3π/2)(180/π) = 270°.

  5. On the unit circle, what are the coordinates for angle 0°?

    • (1, 0)
    • (0, 1)
    • (−1, 0)
    • (0, −1)

    0° is along the positive x-axis, at (1, 0).

  6. On the unit circle, what are the coordinates for angle 90°?

    • (0, 1)
    • (1, 0)
    • (0, −1)
    • (−1, 0)

    90° is straight up the positive y-axis, at (0, 1).

  7. Find sin(30°)

    • 1/2
    • √3/2
    • √2/2
    • 1

    sin(30°) = 1/2, one of the standard unit circle values.

  8. Find cos(60°)

    • 1/2
    • √3/2
    • √2/2
    • 0

    cos(60°) = 1/2.

  9. Find cos(45°)

    • √2/2
    • 1/2
    • √3/2
    • 1

    cos(45°) = √2/2.

  10. Find tan(45°)

    • 1
    • 0
    • Undefined
    • √2

    tan(45°) = sin(45°)/cos(45°) = (√2/2)/(√2/2) = 1.

  11. What is the reference angle for 150°?

    • 30°
    • 150°
    • 60°
    • 180°

    180° − 150° = 30°.

  12. In which quadrant is the angle 200°?

    • III
    • I
    • II
    • IV

    200° is between 180° and 270°, which is Quadrant III.

  13. Using ASTC, which trig function is positive in Quadrant III?

    • Tangent
    • Sine
    • Cosine
    • All of them

    ASTC: Quadrant III is where only Tangent is positive.

  14. Convert 4π/3 radians to degrees

    • 240°
    • 120°
    • 300°
    • 60°

    (4π/3)(180/π) = 240°.

  15. Convert 315° to radians

    • 7π/4
    • 5π/4
    • 3π/2
    • 9π/4

    315(π/180) = 315π/180 = 7π/4.

  16. Find sin(150°)

    • 1/2
    • −1/2
    • √3/2
    • −√3/2

    Reference angle 30°, sine is positive in Quadrant II: sin(150°)=1/2.

  17. Find cos(120°)

    • −1/2
    • 1/2
    • −√3/2
    • √3/2

    Reference angle 60°, cosine is negative in Quadrant II: cos(120°)=−1/2.

  18. Find sin(270°)

    • −1
    • 1
    • 0
    • Undefined

    270° is straight down the unit circle at (0,−1), so sin(270°) = −1.

  19. What is the reference angle for 305°?

    • 55°
    • 305°
    • 125°
    • 35°

    360° − 305° = 55°.

  20. In which quadrant does an angle of 5π/6 radians terminate?

    • II
    • I
    • III
    • IV

    5π/6 = 150°, which is in Quadrant II.

Unit 7: Trigonometric Graphs & Identities (20)

  1. Find the amplitude of y = 3 sin(x)

    • 3
    • 1
    • 2π
    • x

    The amplitude is the coefficient in front of sine: |3| = 3.

  2. Find the period of y = sin(2x)

    • π
    • 2π
    • 4π
    • π/2

    Period = 2π/B = 2π/2 = π.

  3. Find the period of y = cos(x/2)

    • 4π
    • π
    • 2π
    • π/2

    Period = 2π/(1/2) = 4π.

  4. Find the amplitude and midline of y = 2sin(x) + 5

    • Amplitude 2, midline y = 5
    • Amplitude 5, midline y = 2
    • Amplitude 2, midline y = 0
    • Amplitude 7, midline y = 5

    The coefficient 2 is the amplitude; the +5 shifts the midline to y=5.

  5. If sin θ = 3/5, find cos²θ using the Pythagorean identity

    • 16/25
    • 9/25
    • 4/5
    • 1/25

    sin²θ+cos²θ=1 → (3/5)²+cos²θ=1 → cos²θ = 1 − 9/25 = 16/25.

  6. Simplify sec²θ − tan²θ

    • 1
    • 0
    • tan²θ
    • sec²θ

    Since 1+tan²θ=sec²θ, rearranging gives sec²θ−tan²θ=1.

  7. Rewrite tan θ using sine and cosine

    • sin θ / cos θ
    • cos θ / sin θ
    • 1/sin θ
    • 1/cos θ

    The quotient identity: tan θ = sin θ / cos θ.

  8. What is csc θ equivalent to?

    • 1/sin θ
    • 1/cos θ
    • 1/tan θ
    • sin θ

    csc θ is the reciprocal of sin θ.

  9. Solve sin θ = 1/2 for θ in [0°, 360°)

    • 30° or 150°
    • 30° only
    • 30° or 210°
    • 60° or 120°

    Reference angle 30°; sine is positive in Quadrants I and II: 30° and 150°.

  10. Solve cos θ = −1/2 for θ in [0°, 360°)

    • 120° or 240°
    • 60° or 300°
    • 120° only
    • 180°

    Reference angle 60°; cosine is negative in Quadrants II and III: 120° and 240°.

  11. How many solutions does sin θ = 0.5 typically have in one full rotation [0°, 360°)?

    • 2
    • 1
    • 4
    • 0

    A sine equation typically has two solutions per rotation, one in each of two quadrants.

  12. tan θ and sec θ are both undefined whenever which condition is true?

    • cos θ = 0
    • sin θ = 0
    • tan θ = 0
    • θ = 0°

    Both tan θ and sec θ have cos θ in their denominator (directly or via sin θ/cos θ).

  13. What is the range of y = cos(x)?

    • [−1, 1]
    • [0, 1]
    • (−∞, ∞)
    • [−2, 2]

    Cosine oscillates between −1 and 1 for all real x.

  14. Find the period of y = tan(x)

    • π
    • 2π
    • π/2
    • 4π

    The tangent function repeats every π radians, unlike sine and cosine which repeat every 2π.

  15. If cos θ = 5/13 and θ is in Quadrant I, find sin θ

    • 12/13
    • 5/13
    • 13/5
    • 8/13

    sin²θ = 1−(5/13)² = 1−25/169 = 144/169, so sin θ = 12/13 (positive in QI).

  16. Simplify: (1 − sin²θ)

    • cos²θ
    • sin²θ
    • 1
    • −cos²θ

    From the Pythagorean identity, 1 − sin²θ = cos²θ.

  17. Find the phase shift of y = sin(x − π/2)

    • π/2 to the right
    • π/2 to the left
    • π to the right
    • No phase shift

    y=sin(x−C) shifts right by C; here C=π/2.

  18. Solve tan θ = 1 for θ in [0°, 360°)

    • 45° or 225°
    • 45° only
    • 45° or 315°
    • 90° or 270°

    Reference angle 45°; tangent is positive in Quadrants I and III: 45° and 225°.

  19. What is the midline of y = 4cos(x) − 3?

    • y = −3
    • y = 4
    • y = 3
    • y = 0

    The vertical shift (−3) sets the new midline.

  20. cot θ is equivalent to which ratio?

    • cos θ / sin θ
    • sin θ / cos θ
    • 1/sin θ
    • 1/cos θ

    cot θ is the reciprocal of tan θ = sin θ/cos θ, so cot θ = cos θ/sin θ.

Unit 8: Complex Numbers & Quadratics Revisited (20)

  1. Simplify √(−16)

    • 4i
    • −4i
    • 16i
    • −16

    √(−16) = √16 · i = 4i.

  2. Simplify i⁶

    • −1
    • 1
    • i
    • −i

    Powers of i cycle every 4: i⁶ = i^(4+2) = i² = −1.

  3. Simplify i¹⁵

    • −i
    • i
    • −1
    • 1

    15 ÷ 4 leaves remainder 3, so i¹⁵ = i³ = −i.

  4. Add: (3 + 2i) + (5 − 4i)

    • 8 − 2i
    • 8 + 2i
    • −2 + 8i
    • 2 − 8i

    Combine real and imaginary parts separately: (3+5) + (2−4)i = 8 − 2i.

  5. Subtract: (7 − 3i) − (2 + 5i)

    • 5 − 8i
    • 5 + 8i
    • 9 − 8i
    • 5 − 2i

    (7−2) + (−3−5)i = 5 − 8i.

  6. Multiply: (2 + i)(3 − i)

    • 7 + i
    • 7 − i
    • 5 + i
    • 6 − i²

    FOIL: 6 − 2i + 3i − i² = 6 + i − (−1) = 7 + i.

  7. Find the conjugate of 4 − 7i

    • 4 + 7i
    • −4 − 7i
    • −4 + 7i
    • 7 − 4i

    The conjugate of a−bi is a+bi.

  8. Multiply (3 + 2i)(3 − 2i)

    • 13
    • 5
    • 9 − 4i
    • 13 − 12i

    A number times its conjugate: 3² + 2² = 9 + 4 = 13.

  9. Solve x² + 9 = 0

    • x = ±3i
    • x = ±9i
    • x = ±3
    • No solution

    x² = −9, so x = ±√(−9) = ±3i.

  10. Solve x² − 4x + 13 = 0 using the quadratic formula

    • 2 ± 3i
    • 4 ± 6i
    • 2 ± 6i
    • −2 ± 3i

    Discriminant = 16−52 = −36. x = (4±√(−36))/2 = (4±6i)/2 = 2±3i.

  11. A quadratic with real coefficients has one solution 5 − 2i. What must the other solution be?

    • 5 + 2i
    • −5 − 2i
    • −5 + 2i
    • 5 − 2i

    Complex solutions to a real-coefficient quadratic always come in conjugate pairs.

  12. Divide: 4/(1 + i)

    • 2 − 2i
    • 4 − 4i
    • 2 + 2i
    • 4(1−i)

    Multiply by the conjugate (1−i)/(1−i): 4(1−i)/2 = 2−2i.

  13. What is the discriminant of x² + 2x + 5 = 0, and what does it indicate?

    • −16; two complex conjugate solutions
    • 16; two real solutions
    • −16; no solutions exist
    • 4; one real solution

    Discriminant = 4 − 20 = −16, which is negative, meaning two complex conjugate solutions.

  14. Simplify: i² + i⁴

    • 0
    • −1
    • 1
    • 2

    i²=−1 and i⁴=1, so −1+1=0.

  15. Simplify √(−25) · √(−4)

    • −10
    • 10
    • 10i
    • −10i

    √(−25)=5i and √(−4)=2i, so 5i·2i=10i²=10(−1)=−10.

  16. Multiply: (1 + i)²

    • 2i
    • 2
    • 1 + 2i
    • 1 − 2i

    (1+i)(1+i) = 1+i+i+i² = 1+2i−1 = 2i.

  17. Solve 2x² + 8 = 0

    • x = ±2i
    • x = ±4i
    • x = ±2
    • No real or complex solution

    x² = −4, so x = ±√(−4) = ±2i.

  18. Find the sum of the complex conjugate pair 3+5i and 3−5i

    • 6
    • 10i
    • 6+10i
    • 0

    (3+5i)+(3−5i) = 6 (the imaginary parts cancel).

  19. Simplify: (2−3i) − (−1+4i)

    • 3 − 7i
    • 1 + i
    • 1 − 7i
    • 3 + i

    (2−(−1)) + (−3−4)i = 3 − 7i.

  20. For x² − 6x + 25 = 0, find the discriminant and describe the solutions

    • −64; two complex conjugate solutions
    • 64; two real solutions
    • −64; one real solution
    • 36; two real solutions

    Discriminant = 36 − 100 = −64, negative, so two complex conjugate solutions.

Unit 9: Function Operations, Inverses & Transformations (20)

  1. If f(x) = x+3 and g(x) = x², find (f+g)(x)

    • x² + x + 3
    • x² + x + 9
    • x³ + 3
    • x² + 3

    (f+g)(x) = f(x)+g(x) = (x+3)+x² = x²+x+3.

  2. If f(x) = 2x and g(x) = x−1, find (f·g)(x)

    • 2x² − 2x
    • 2x − 2
    • 2x² − 1
    • x² − 2x

    f(x)·g(x) = 2x(x−1) = 2x²−2x.

  3. If f(x) = x² and g(x) = x+1, find (f∘g)(x)

    • x² + 2x + 1
    • x² + 1
    • x² + x + 1
    • 2x + 1

    f(g(x)) = f(x+1) = (x+1)² = x²+2x+1.

  4. If f(x) = x+1 and g(x) = x², find (g∘f)(x)

    • x² + 2x + 1
    • x² + 1
    • x² + 2x
    • 2x + 1

    g(f(x)) = g(x+1) = (x+1)² = x²+2x+1.

  5. If f(x) = 2x − 4, find f⁻¹(x)

    • (x+4)/2
    • 2x+4
    • (x−4)/2
    • x/2+4

    Swap x and y: x=2y−4 → x+4=2y → y=(x+4)/2.

  6. If f(x) = 3x + 6, find f⁻¹(2)

    • −4/3
    • −2
    • 4/3
    • 8/3

    f⁻¹(x)=(x−6)/3, so f⁻¹(2)=(2−6)/3=−4/3.

  7. If f(5) = 12, what must be true about f⁻¹?

    • f⁻¹(12) = 5
    • f⁻¹(5) = 12
    • f⁻¹(12) = 1/5
    • f⁻¹(5) = 1/12

    An inverse function swaps inputs and outputs, so f⁻¹(12)=5.

  8. Which test determines whether a function's inverse is also a function?

    • Horizontal Line Test
    • Vertical Line Test
    • Slope Test
    • Zero Test

    If a horizontal line crosses the graph more than once, the inverse fails to be a function.

  9. The graphs of f and f⁻¹ are reflections of each other across which line?

    • y = x
    • The x-axis
    • The y-axis
    • y = −x

    Swapping x and y to find an inverse corresponds to reflecting across y=x.

  10. If g(x) = f(x−3) + 2, how is the graph of g related to f?

    • Shifted right 3, up 2
    • Shifted left 3, up 2
    • Shifted right 3, down 2
    • Shifted left 3, down 2

    f(x−3) shifts right 3; adding 2 outside shifts up 2.

  11. If g(x) = −f(x), how is the graph of g related to f?

    • Reflected over the x-axis
    • Reflected over the y-axis
    • Shifted down
    • Shifted up

    Negating the whole output flips the graph vertically, over the x-axis.

  12. If g(x) = f(−x), how is the graph of g related to f?

    • Reflected over the y-axis
    • Reflected over the x-axis
    • Shifted left
    • Shifted right

    Negating the input flips the graph horizontally, over the y-axis.

  13. Does f(x) = x³ have an inverse that is also a function?

    • Yes, it passes the Horizontal Line Test
    • No, cubics never have inverses
    • Only if restricted to x ≥ 0
    • No, only quadratics have inverses

    x³ is one-to-one (strictly increasing), so it passes the Horizontal Line Test and has an inverse function.

  14. If f(x) = √x and g(x) = x+4, find the domain of (f∘g)(x)

    • x ≥ −4
    • x ≥ 0
    • x ≥ 4
    • All real numbers

    f(g(x)) = √(x+4), which requires x+4 ≥ 0, so x ≥ −4.

  15. If f(x) = x² (for x ≥ 0) and g(x) = √x, are f and g inverses of each other?

    • Yes, f(g(x))=x and g(f(x))=x for x≥0
    • No, they are not related
    • Only f(g(x))=x works
    • Only g(f(x))=x works

    f(g(x)) = (√x)² = x and g(f(x)) = √(x²) = x for x≥0 — both checks pass.

  16. If h(x) = f(g(x)) and g(x) = 2x, f(x) = x−5, find h(3)

    • 1
    • 6
    • −4
    • 11

    g(3)=6, then f(6)=6−5=1.

  17. If f(x) = 1/x, find f⁻¹(x)

    • 1/x (it's its own inverse)
    • x
    • −1/x
    • x²

    Swapping x and y in y=1/x gives x=1/y, so y=1/x — the same function.

  18. If g(x) = 2f(x), how does the graph of g compare to f?

    • Vertically stretched by a factor of 2
    • Horizontally stretched by a factor of 2
    • Shifted up 2 units
    • Shifted right 2 units

    Multiplying the output by 2 stretches the graph vertically by a factor of 2.

  19. If g(x) = f(2x), how does the graph of g compare to f?

    • Horizontally compressed by a factor of 2
    • Horizontally stretched by a factor of 2
    • Vertically compressed by a factor of 2
    • Shifted left 2 units

    Multiplying the input by 2 compresses the graph horizontally by a factor of 2.

  20. If f(x) = x+7 and g(x) = x−7, what is (f∘g)(x)?

    • x
    • x+14
    • x−14
    • 2x

    f(g(x)) = f(x−7) = (x−7)+7 = x — confirming f and g are inverses.

Unit 10: Statistics: Sampling & Inference (20)

  1. Which sampling method gives every member of the population an equal chance of selection?

    • Simple random sample
    • Convenience sample
    • Voluntary response sample
    • Judgment sample

    That's the definition of a simple random sample.

  2. A survey only asks people leaving a gym about exercise habits. What kind of sample is this?

    • A biased convenience sample
    • A simple random sample
    • A stratified sample
    • An unbiased sample

    Only surveying gym-goers is a convenience sample that's biased toward people who already exercise.

  3. According to the Empirical Rule, about what percent of normally distributed data falls within 1 standard deviation of the mean?

    • 68%
    • 95%
    • 99.7%
    • 50%

    The 68-95-99.7 Rule: about 68% falls within 1 standard deviation.

  4. According to the Empirical Rule, about what percent falls within 2 standard deviations?

    • 95%
    • 68%
    • 99.7%
    • 90%

    About 95% falls within 2 standard deviations.

  5. A data set has mean 100 and standard deviation 15. Find the z-score for x = 130

    • 2
    • 1
    • 3
    • 0.5

    z = (130−100)/15 = 30/15 = 2.

  6. A data set has mean 50 and standard deviation 8. Find the z-score for x = 42

    • −1
    • 1
    • −8
    • 8

    z = (42−50)/8 = −8/8 = −1.

  7. A z-score of −2 means the value is:

    • 2 standard deviations below the mean
    • 2 standard deviations above the mean
    • 2 units below zero
    • The mean minus 2

    A negative z-score means the value is below the mean, by that many standard deviations.

  8. As sample size increases, what generally happens to the margin of error?

    • It decreases
    • It increases
    • It stays the same
    • It becomes zero

    Larger samples generally produce more precise estimates, shrinking the margin of error.

  9. A 95% confidence interval means:

    • About 95% of intervals built this way would contain the true population value
    • There's a 95% chance this specific interval is correct
    • 95% of the data falls in this interval
    • The sample is 95% accurate

    Confidence level describes the long-run reliability of the METHOD, not one specific interval.

  10. A confidence interval is calculated as:

    • Sample statistic ± margin of error
    • Sample statistic × margin of error
    • Population mean ± z-score
    • Mean − standard deviation

    A confidence interval is the sample statistic plus or minus the margin of error.

  11. A larger sample size directly fixes which problem?

    • High sampling variability (reduces margin of error)
    • A biased sampling method
    • An incorrect hypothesis
    • Missing data

    Sample size affects precision (variability), but does NOT fix a biased collection method.

  12. What is a stratified sample?

    • The population is divided into subgroups, then randomly sampled from each proportionally
    • Only volunteers are surveyed
    • Every 10th person is surveyed
    • Only the largest subgroup is surveyed

    Stratified sampling ensures proportional representation from each subgroup.

  13. A data value has z-score 0. What does this mean?

    • It equals the mean exactly
    • It is the smallest value in the data set
    • It is an outlier
    • It is undefined

    A z-score of 0 means the value is exactly at the mean (zero standard deviations away).

  14. Which of these is an example of a stratified sample?

    • Randomly sampling proportionally from each grade level in a school
    • Surveying only the first 20 students who arrive
    • Asking for volunteers over social media
    • Surveying every student in one classroom

    Stratified sampling divides the population into subgroups (like grade levels) then samples proportionally from each.

  15. A data set is normally distributed with mean 60 and standard deviation 5. About what percent of data falls between 55 and 65?

    • 68%
    • 95%
    • 50%
    • 99.7%

    55 to 65 is exactly ±1 standard deviation from the mean — about 68% by the Empirical Rule.

  16. A data set has mean 200 and standard deviation 25. Find the value corresponding to z = 1.5

    • 237.5
    • 225
    • 250
    • 212.5

    x = μ + zσ = 200 + 1.5(25) = 200+37.5 = 237.5.

  17. Which of the following would most likely increase bias in a survey?

    • Only surveying people who choose to respond online
    • Randomly selecting participants from the full population
    • Increasing the sample size with random selection
    • Using a larger, still-random sample

    Voluntary/self-selected responses are a classic source of bias, regardless of sample size.

  18. A result from a simulation falls far outside the range typically produced by random chance. What does this suggest?

    • The result is likely NOT due to random chance alone
    • The simulation was run incorrectly
    • The result must be an error
    • Nothing can be concluded

    An unusually extreme result compared to the simulated random-chance distribution is evidence against pure chance.

  19. What happens to a confidence interval's width if you increase the confidence level (e.g., from 90% to 99%)?

    • The interval gets wider
    • The interval gets narrower
    • The interval stays the same width
    • It becomes a single point

    Higher confidence requires a wider range to be more sure of capturing the true value.

  20. Which best describes a census, as opposed to a sample?

    • Collecting data from every member of the population
    • Collecting data from a random subset
    • Collecting data only from volunteers
    • Estimating data using a formula

    A census surveys the ENTIRE population, not just a sample of it.

Unit 11: Probability (20)

  1. A bag has 4 red and 6 blue marbles. Find P(red)

    • 2/5
    • 4/6
    • 6/10
    • 1/4

    P(red) = 4/10 = 2/5.

  2. A die is rolled. Find P(rolling a 4 or a 6)

    • 1/3
    • 1/6
    • 2/3
    • 1/2

    Mutually exclusive outcomes: 1/6 + 1/6 = 2/6 = 1/3.

  3. A card is drawn from a standard deck. Find P(king or queen)

    • 2/13
    • 1/13
    • 4/13
    • 1/26

    4 kings + 4 queens out of 52: 8/52 = 2/13.

  4. Two coins are flipped. Find P(both heads)

    • 1/4
    • 1/2
    • 1/3
    • 3/4

    Independent events: (1/2)(1/2) = 1/4.

  5. A bag has 3 red and 2 blue marbles. Two are drawn WITHOUT replacement. Find P(both red)

    • 3/10
    • 9/25
    • 3/5
    • 1/5

    (3/5)(2/4) = 6/20 = 3/10 — the second draw's probability changes since it's dependent.

  6. In a class of 10 boys and 15 girls, one student is chosen at random. Find P(girl)

    • 3/5
    • 15/10
    • 2/5
    • 1/2

    P(girl) = 15/25 = 3/5.

  7. Events A and B are independent, with P(A) = 0.4 and P(B) = 0.5. Find P(A and B)

    • 0.20
    • 0.90
    • 0.10
    • 0.45

    For independent events: P(A and B) = P(A)·P(B) = 0.4 × 0.5 = 0.20.

  8. How many ways can 3 of 6 people be arranged in a line (order matters)?

    • 120
    • 20
    • 216
    • 720

    6P3 = 6!/(6−3)! = 6·5·4 = 120.

  9. How many ways can a committee of 3 be chosen from 6 people (order doesn't matter)?

    • 20
    • 120
    • 216
    • 18

    6C3 = 6!/(3!3!) = 720/36 = 20.

  10. A password uses 4 different letters, in a specific order, chosen from 10 available letters. How many are possible?

    • 5040
    • 210
    • 10000
    • 151200

    10P4 = 10·9·8·7 = 5040.

  11. A committee of 2 is chosen from 5 people. How many different committees are possible?

    • 10
    • 20
    • 25
    • 120

    5C2 = 5!/(2!3!) = 120/12 = 10.

  12. A fair coin is flipped 4 times. Find P(exactly 2 heads)

    • 0.375
    • 0.25
    • 0.5
    • 0.0625

    4C2·(0.5)²·(0.5)² = 6·0.25·0.25 = 0.375.

  13. P(A) = 0.3 and events A and B are mutually exclusive with P(B) = 0.2. Find P(A or B)

    • 0.5
    • 0.06
    • 0.1
    • 0.44

    Mutually exclusive events have no overlap: P(A or B) = P(A)+P(B) = 0.3+0.2 = 0.5.

  14. A spinner has 8 equal sections numbered 1–8. Find P(spinning an even number)

    • 1/2
    • 1/4
    • 3/8
    • 5/8

    4 even numbers (2,4,6,8) out of 8 total: 4/8 = 1/2.

  15. Two dice are rolled. Find P(sum equals 7)

    • 1/6
    • 1/12
    • 1/36
    • 1/9

    6 combinations sum to 7 out of 36 total: 6/36 = 1/6.

  16. A box has 5 red, 3 blue, and 2 green marbles. Find P(blue or green)

    • 1/2
    • 3/10
    • 1/5
    • 4/5

    (3+2)/10 = 5/10 = 1/2.

  17. If P(A) = 0.6, find P(not A)

    • 0.4
    • 0.6
    • 1.6
    • 0

    P(not A) = 1 − P(A) = 1 − 0.6 = 0.4.

  18. How many different 4-digit codes can be made using digits 0–9 if repetition IS allowed?

    • 10,000
    • 5,040
    • 210
    • 40

    Each of the 4 positions has 10 choices: 10⁴ = 10,000.

  19. A fair coin is flipped 5 times. Find P(exactly 0 heads)

    • 1/32
    • 1/2
    • 5/32
    • 0

    5C0·(0.5)⁰·(0.5)⁵ = 1·1·(1/32) = 1/32.

  20. In a group of 8 people, how many different pairs (order doesn't matter) can be formed?

    • 28
    • 56
    • 64
    • 16

    8C2 = 8!/(2!6!) = 56/2 = 28.

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Unit 1: Polynomial Functions & Operations

Remainder Theorem
If P(x) is divided by (x − c), the remainder equals P(c).
Factor Theorem
(x − c) is a factor of P(x) if and only if P(c) = 0.
Synthetic division
A fast shortcut for dividing a polynomial by a linear factor (x − c).
Multiplicity
How many times a zero's factor repeats. Odd multiplicity crosses the x-axis; even multiplicity bounces off it.
End behavior
How a polynomial's graph behaves as x → ±∞, controlled by the degree and the leading coefficient's sign.

Unit 2: Rational Expressions & Equations

Excluded value
Any x-value that makes a rational expression's original denominator equal zero — always undefined.
Rational expression
A fraction with polynomials in the numerator and denominator.
Least Common Denominator (LCD)
The smallest expression that all denominators divide into evenly; found by factoring each denominator.
Extraneous solution
A solution produced algebraically that doesn't actually work in the original equation — common after clearing denominators.

Unit 3: Radicals & Rational Exponents

Like radicals
Radicals with the same index and the same expression underneath — only these can be added or subtracted directly.
Rationalizing the denominator
Eliminating a radical from a denominator by multiplying by an appropriate radical or conjugate.
Conjugate (radical)
For a binomial like a + √b, its conjugate is a − √b — multiplying them eliminates the radical.
x^(m/n)
Equals ⁿ√(xᵐ) — the denominator n is the root, the numerator m is the power.

Unit 4: Exponential & Logarithmic Functions

Logarithm
logₐ(x) = y means aʸ = x — a log answers 'what exponent is needed?'
Common log
log(x) with no base written means base 10.
Natural log
ln(x) means logₑ(x), using base e ≈ 2.718.
Product Rule (logs)
logₐ(MN) = logₐM + logₐN.
Quotient Rule (logs)
logₐ(M/N) = logₐM − logₐN.
Power Rule (logs)
logₐ(Mᵖ) = p · logₐM.

Unit 5: Sequences & Series

Sigma notation
Compact notation (Σ) for writing the sum of a sequence of terms.
Arithmetic series formula
Sₙ = n/2 · (a₁ + aₙ) — n times the average of the first and last term.
Geometric series formula (finite)
Sₙ = a₁(1 − rⁿ)/(1 − r), for r ≠ 1.
Infinite geometric series
S = a₁/(1 − r), but only converges (has a sum) when |r| &lt; 1.

Unit 6: Trigonometric Functions & the Unit Circle

Radian
An angle measure based on the circle's radius; 2π radians = one full revolution (360°).
Unit circle
A circle of radius 1 centered at the origin, used to define trig functions for every angle.
Reference angle
The acute angle (0°–90°) between an angle's terminal side and the x-axis.
ASTC
'All Students Take Calculus' — a mnemonic for which trig functions are positive in each quadrant (All, Sin, Tan, Cos).

Unit 7: Trigonometric Graphs & Identities

Amplitude
|A| in y = A sin(Bx−C)+D — the height of the wave from the midline to its peak.
Period
2π/B in y = A sin(Bx−C)+D — the horizontal length of one full wave cycle.
Pythagorean identity
sin²θ + cos²θ = 1, true for every angle θ.

Unit 8: Complex Numbers & Quadratics Revisited

Imaginary unit i
Defined so that i² = −1, allowing negative numbers to have square roots.
Complex number
A number of the form a + bi, with a real part a and an imaginary part b.
Complex conjugate
For a + bi, the conjugate is a − bi. Multiplying them gives the real number a² + b².
Powers of i cycle
i¹=i, i²=−1, i³=−i, i⁴=1 — then the pattern repeats every 4 powers.

Unit 9: Function Operations, Inverses & Transformations

Composition of functions
(f∘g)(x) = f(g(x)) — evaluate g first, then plug that result into f.
Inverse function
f⁻¹(x) undoes f(x): f(f⁻¹(x)) = x and f⁻¹(f(x)) = x.
Horizontal Line Test
If any horizontal line crosses a graph more than once, its inverse is not a function.

Unit 10: Statistics: Sampling & Inference

Simple random sample
A sample where every member of the population has an equal chance of being chosen.
Empirical Rule
For a normal distribution: 68% of data within 1 standard deviation, 95% within 2, 99.7% within 3.
z-score
z = (x − μ) / σ — how many standard deviations a value is from the mean.
Margin of error
Accounts for natural sampling variability; generally shrinks as sample size grows.
Confidence interval
sample statistic ± margin of error — a range likely to contain the true population value.

Unit 11: Probability

Independent events
Events where one outcome does not affect the other's probability: P(A and B) = P(A)·P(B).
Conditional probability
P(B | A): the probability of B, given that A has already happened.
Permutation
An arrangement where ORDER matters. nPr = n!/(n−r)!.
Combination
A selection where order does NOT matter. nCr = n!/(r!(n−r)!).
Binomial probability
P(exactly k successes in n trials) = nCk · pᵏ · (1−p)^(n−k).

Core Formulas

Remainder Theorem
P(x)÷(x−c) has remainder P(c)
Factor Theorem
(x−c) is a factor ⟺ P(c)=0
Rational Exponent
x^(m/n) = ⁿ√(xᵐ)
Logarithm Definition
logₐ(x)=y ⟺ aʸ=x
Product Rule (logs)
logₐ(MN)=logₐM+logₐN
Quotient Rule (logs)
logₐ(M/N)=logₐM−logₐN
Power Rule (logs)
logₐ(Mᵖ)=p·logₐM
Arithmetic Series
Sₙ = n/2·(a₁+aₙ)
Geometric Series (finite)
Sₙ = a₁(1−rⁿ)/(1−r)
Geometric Series (infinite)
S = a₁/(1−r), needs |r|<1
Radians ↔ Degrees
×π/180 or ×180/π
Pythagorean Identity
sin²θ + cos²θ = 1
Amplitude / Period
|A| / (2π ÷ B) for A sin(Bx−C)+D
Imaginary Unit
i² = −1 · √(−n) = i√n
Complex Conjugate
(a+bi)(a−bi) = a²+b²
Composition
(f∘g)(x) = f(g(x))
z-score
z = (x−μ)/σ
Empirical Rule
68% / 95% / 99.7% within 1/2/3σ
Permutation
nPr = n!/(n−r)!
Combination
nCr = n!/(r!(n−r)!)
Binomial Probability
nCk·pᵏ·(1−p)ⁿ⁻ᵏ

Unit Circle: Key Angles

DegreesRadianssincostan
0°0010
30°π/61/2√3/2√3/3
45°π/4√2/2√2/21
60°π/3√3/21/2√3
90°π/210undefined
180°π0−10
270°3π/2−10undefined

Powers of i

PowerValue
i¹i
i²−1
i³−i
i⁴1
i⁵, i⁹, i¹³...i (cycle repeats)

Fast Facts

  • ASTC: Quadrant I all positive, II sine only, III tangent only, IV cosine only.
  • An infinite geometric series only has a sum when |r| < 1.
  • logₐ(M+N) is NOT the same as logₐM + logₐN — the product rule only works for multiplication.
  • Complex solutions to a real-coefficient quadratic always come in conjugate pairs.
  • Order matters for permutations; it doesn't for combinations.
  • Always check units and excluded values in your final answer before moving on.
Quick ways to lock in the facts you keep forgetting. Read the big trick, then the small note tells you what it unlocks. Say them out loud — silly is memorable.

Polynomials & Rational Expressions

P(c) is the remainder
Plugging c into a polynomial gives you the exact same number as the remainder from dividing by (x−c) — no need to do both.
Cancel factors, not terms
You can only cancel things that are MULTIPLIED together, never things being added or subtracted — factor first, always.

Radicals & Logs

Logs turn multiplication into addition
log(MN) = log M + log N. Logs are built to turn hard multiplication problems into easy addition problems — that's their whole purpose historically.
A log can't eat a negative
You can never take the log of zero or a negative number — always check that your final answer keeps every log's input positive.

Sequences & Series

|r| < 1 or it blows up
An infinite geometric series only settles down to a finite sum when |r| is less than 1 — otherwise the terms keep growing forever.

Trigonometry

All Students Take Calculus
ASTC tells you which trig function is positive in each quadrant, going counterclockwise from QI: All, Sine, Tangent, Cosine.
Reference angle: always positive, always acute
No matter which quadrant the real angle is in, its reference angle is always between 0° and 90° — find it, then fix the sign using ASTC.

Complex Numbers

i-squared is negative one
Every complex-number simplification comes down to remembering i² = −1 and substituting it in wherever it appears.
Powers of i repeat every 4
i, −1, −i, 1, then it repeats. To simplify a big power, divide by 4 and use the remainder.

Functions

Composition: inside out
(f∘g)(x) means g happens first, then f wraps around the result — read it right to left, like unwrapping a package from the inside.

Statistics & Probability

Bigger sample, smaller margin
A larger random sample shrinks the margin of error — but it never fixes a biased collection method. Size and bias are two separate problems.
Order matters, or it doesn't
If rearranging the same people/items counts as a different outcome, use a permutation. If it counts as the same outcome, use a combination.
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