Algebra 2 Master Quiz | 20 Essential Problems

Unit 01

📊 Polynomial Functions & Equations

📌 Key Concepts

A polynomial of degree n has at most n real zeros. The Factor Theorem states that (x − c) is a factor of f(x) if and only if f(c) = 0. The Remainder Theorem states that when f(x) is divided by (x − c), the remainder equals f(c).

f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀

Degree n → at most n zeros f(c) = 0 → (x−c) is a factor Remainder = f(c)

▶ Example: If f(x) = x³ − 4x² + x + 6, is (x − 2) a factor?

f(2) = 8 − 16 + 2 + 6 = 0 ✓ → Yes, (x−2) is a factor

Unit 02

📐 Quadratic Equations & Functions

📌 Key Concepts

The Quadratic Formula solves any quadratic. The discriminant (b² − 4ac) determines the nature of roots: positive → 2 real roots; zero → 1 repeated root; negative → 2 complex roots. Vertex form is f(x) = a(x − h)² + k, with vertex (h, k).

x = (−b ± √(b² − 4ac)) / (2a)

Δ > 0 → 2 real roots Δ = 0 → 1 repeated root Δ < 0 → 2 complex roots Vertex: (h, k) in a(x−h)²+k

▶ Example: How many real solutions does 2x² − 3x + 5 = 0 have?

Δ = 9 − 40 = −31 < 0 → 2 complex, no real solutions

Unit 03

🔗 Rational Functions

📌 Key Concepts

A rational function is f(x) = p(x)/q(x). Vertical asymptotes occur where q(x) = 0 (and p(x) ≠ 0). Horizontal asymptotes: if degree of p < degree of q → y = 0; equal degrees → y = leading coefficient ratio; degree of p > degree of q → no horizontal asymptote (oblique instead).

f(x) = p(x) / q(x), q(x) ≠ 0

VA: q(x) = 0 HA: compare degrees of p and q Hole: common factor cancels

▶ Example: Find the HA of f(x) = (3x² + 1)/(x² − 4)

Same degree → HA: y = 3/1 = 3

Unit 04

📈 Exponential & Logarithmic Functions

📌 Key Concepts

Exponential: f(x) = aˣ (a > 0, a ≠ 1). Logarithm is the inverse: logₐ(b) = c ↔ aᶜ = b. Key log laws: Product, Quotient, Power rules. Change of base: logₐ(x) = ln(x)/ln(a).

logₐ(MN) = logₐM + logₐN | logₐ(Mᵖ) = p·logₐM

log(ab) = log a + log b log(a/b) = log a − log b log(aⁿ) = n·log a ln e = 1, log 10 = 1

▶ Example: Simplify log₂(32)

log₂(2⁵) = 5

Unit 05

⭕ Conic Sections

📌 Key Concepts

Circle: (x−h)² + (y−k)² = r². Parabola: y = a(x−h)² + k or x = a(y−k)² + h. Ellipse: x²/a² + y²/b² = 1 (a > b). Hyperbola: x²/a² − y²/b² = 1.

Circle: (x−h)² + (y−k)² = r² | Ellipse: x²/a² + y²/b² = 1

Same sign, equal → circle Same sign, unequal → ellipse Opposite signs → hyperbola

▶ Example: What shape is x² + y² − 6x + 2y = 6?

Complete square: (x−3)² + (y+1)² = 16 → Circle, r = 4

Unit 06

🔢 Sequences & Series

📌 Key Concepts

Arithmetic: aₙ = a₁ + (n−1)d; S = n/2(a₁ + aₙ). Geometric: aₙ = a₁·rⁿ⁻¹; S = a₁(1−rⁿ)/(1−r). Infinite Geometric Series (|r| < 1): S∞ = a₁/(1−r).

Arith: aₙ = a₁ + (n−1)d | Geo: aₙ = a₁·rⁿ⁻¹

Arithmetic: add d each time Geometric: multiply r each time |r| < 1 → infinite sum exists

▶ Example: Sum of infinite geometric series: a₁ = 8, r = 1/2

S = 8/(1 − 1/2) = 8/(1/2) = 16

Unit 07

🎲 Probability & Statistics

📌 Key Concepts

Permutation: P(n,r) = n!/(n−r)! (order matters). Combination: C(n,r) = n!/[r!(n−r)!] (order does not matter). Binomial Theorem: (a+b)ⁿ = Σ C(n,k)·aⁿ⁻ᵏ·bᵏ.

C(n,r) = n! / [r!(n−r)!] | P(n,r) = n! / (n−r)!

Order matters → Permutation Order irrelevant → Combination C(n,0) = C(n,n) = 1

▶ Example: How many ways to choose 3 from 7? (order doesn't matter)

C(7,3) = 7!/(3!·4!) = 35

Unit 08

📉 Systems of Equations & Matrices

📌 Key Concepts

A 2×2 system has a unique solution when the determinant ≠ 0. Determinant of [[a,b],[c,d]] = ad − bc. Cramer's Rule uses determinants to solve systems. Matrix multiplication is NOT commutative: AB ≠ BA in general.

det[[a,b],[c,d]] = ad − bc

det ≠ 0 → unique solution det = 0 → no unique solution AB ≠ BA (not commutative)

▶ Example: det[[3,2],[1,4]] = ?

3·4 − 2·1 = 12 − 2 = 10

Algebra 2

What You Must Know