RegentsLab

Algebra 2 Regents Review Sheet

Alg I's rules + new function families (logs, trig, complex). Master the identities and everything else clicks.

Practice Algebra 2 freeReference & formulas

Big ideas

Polynomials & factoring

A polynomial is a sum of terms with whole-number exponents. Factoring finds its roots.

Rational Root Theorem

Possible rational roots = ±(factors of constant)/(factors of leading coefficient).

Example: x³ − 2x² − 5x + 6 → Test ±1, ±2, ±3, ±6 → x = 1 works
Sum & difference of cubes

a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)

Middle sign in the trinomial flips.

Example: x³ − 8 → (x − 2)(x² + 2x + 4)
Synthetic division

Quick way to divide by (x − c). Bring down leading coefficient, multiply, add.

Example: Divide x² − 3x + 2 by (x − 1) → Quotient x − 2, remainder 0

Trap: Don't forget to check for a GCF first. If every term has a common factor, pull it out before doing anything else.

Tip: End behavior: even degree with positive lead → both ends up. Odd degree with positive lead → left down, right up.

Rational & radical functions

Rational = fraction of polynomials. Radical = has a root. Both love hidden restrictions.

Simplify rational

Factor numerator and denominator, cancel common factors, note excluded values.

Example: (x² − 4)/(x² − x − 2) → = (x−2)(x+2)/((x−2)(x+1)) = (x+2)/(x+1), x ≠ 2, −1
Solve rational equation

Multiply by LCD to clear denominators, then solve. Check for extraneous roots.

Example: 1/x + 1/2 = 1 → Multiply by 2x: 2 + x = 2x → x = 2 (check: not excluded)
Radical equation

Isolate the radical, square both sides. ALWAYS check answers — squaring can create phantom solutions.

Example: √(x + 6) = x → Square: x + 6 = x² → x² − x − 6 = 0 → x = 3 (x = −2 rejected)

Trap: Squaring introduces extraneous solutions. Always plug your answer back into the ORIGINAL equation.

Tip: Vertical asymptotes come from denominator zeros that DIDN'T cancel. Holes come from ones that did.

Exponents & logarithms

Logs are exponents in disguise. log_b(x) = y means b^y = x.

Log ↔ exponent

log_b(x) = y ⟺ b^y = x.

Example: log_2(8) = ? → 2^y = 8 → y = 3
Log properties

\log(ab) = \log a + \log b,\ \log(a/b) = \log a - \log b,\ \log(a^n) = n\log a

Multiplication → addition. Division → subtraction. Power → coefficient.

Example: log(x²y) = ? → 2 log x + log y
Change of base

\log_b(x) = \ln(x)/\ln(b)

When your calculator only has ln or log₁₀.

Example: log_2(50) on a calculator → ln(50)/ln(2) ≈ 5.64
Solve exponentials

Take log of both sides. Bring exponent down as a coefficient.

Example: 3^x = 20 → x = log 20 / log 3 ≈ 2.73

Trap: log(a + b) is NOT log a + log b. Log properties only work on products, quotients, and powers.

Tip: e ≈ 2.718. ln(e) = 1, ln(1) = 0. Continuous growth uses e: A = Pe^(rt).

Trigonometry

The unit circle is your one-stop shop: coordinates (cos θ, sin θ) at every angle.

Six ratios

sin = opposite/hypotenuse, cos = adjacent/hyp, tan = opposite/adjacent. csc/sec/cot are the reciprocals.

Example: sin 30° = ? → 1/2
Radians & degrees

\pi\ \text{radians} = 180°

Multiply by π/180 to convert degrees → radians.

Example: 60° in radians → 60 × π/180 = π/3
Pythagorean identity

\sin^2\theta + \cos^2\theta = 1

Divide by cos² to get 1 + tan² = sec². Divide by sin² to get 1 + cot² = csc².

Example: sin θ = 3/5, cos θ? → cos²θ = 1 − 9/25 = 16/25 → cos θ = ±4/5
Solving trig equations

Isolate the trig function, use inverse. Then find ALL angles in the interval (add 2π, or use the reference angle in each quadrant).

Example: sin θ = 1/2, 0 ≤ θ < 2π → θ = π/6 or 5π/6

Trap: Calculator in the wrong mode. Radian vs degree — check every problem.

Tip: Memorize the unit circle values for 0, π/6, π/4, π/3, π/2 in each quadrant. Everything else is a reference-angle move.

Sequences & series

Sequence = ordered list. Series = sum of a sequence.

Arithmetic

a_n = a_1 + (n-1)d

Adds d each step. d is the common difference.

Example: a₁ = 3, d = 4. Find a₅ → 3 + 4(4) = 19
Geometric

a_n = a_1 \cdot r^{n-1}

Multiplies by r each step. r is the common ratio.

Example: a₁ = 5, r = 2. Find a₄ → 5 × 2³ = 40
Arithmetic series

S_n = (n/2)(a_1 + a_n)

Average the first and last, times how many.

Example: Sum 1 + 2 + ... + 100 → 50(1 + 100) = 5050
Geometric series

S_n = a_1(1 - r^n)/(1 - r)

For finite sums. If |r| < 1, S_∞ = a₁/(1 − r).

Example: Sum of 2 + 4 + 8 + 16 → 2(1 − 2⁴)/(1 − 2) = 30

Trap: n starts at 1. If asked for the 5th term, use n = 5, not 4.

Tip: Regents loves 'a bank account with 5% interest' problems — that's geometric growth with r = 1.05.

Statistics & probability

Normal distributions, simulations, and the difference between correlation and causation.

Normal distribution rule

68% of data within 1 sd of mean. 95% within 2 sd. 99.7% within 3.

Example: Mean 70, sd 5. What % is between 60 and 80? → That's ±2 sd → 95%
Z-score

z = (x - \mu)/\sigma

How many standard deviations from the mean.

Example: μ = 100, σ = 15. Find z for x = 130 → z = 30/15 = 2
Conditional probability

P(A|B) = P(A \cap B)/P(B)

'Probability of A given B.' Restrict the sample space to just the B outcomes.

Example: P(diamond | red card in a deck) → 13/26 = 1/2
Simulation

Regents uses randint on a calculator to simulate. Do many trials, count successes, divide.

Example: Estimate P of at least one 6 in 4 dice rolls → Simulate 100+ trials, count successes/total

Trap: Independent vs mutually exclusive. Independent: P(A and B) = P(A)P(B). Mutually exclusive: P(A and B) = 0.

Tip: For normal distribution problems, sketch the bell curve, shade what's asked, then decide which 68/95/99.7 slice you're in.

Frequently asked questions

How should I study for the Algebra 2 Regents?

Work real past-exam questions by topic, review this one-page summary of the must-know concepts, and read the worked solution every time you miss one. The exam reuses the same ideas year after year.

More Algebra 2 help