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Algebra 2 · All Units · 20 Expert Questions
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Hi! I'm Rex! 🦕 Let's crush Algebra 2 together! Good luck on all 20 questions!
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🌿 Unit 1 — Polynomials Q 1–2
📘 Key Concept
A polynomial is a sum of terms of the form aₙxⁿ. The degree is the highest exponent. Division of polynomials can be done by long division or synthetic division.
Remainder Theorem: f(c) = remainder when f(x) ÷ (x − c)
Factor Theorem: (x − c) is a factor ⟺ f(c) = 0
📝 Example
Find the remainder when f(x) = x³ − 4x + 6 is divided by (x − 2):
f(2) = 8 − 8 + 6 = 6 ✓
Q 1Polynomials
What is the remainder when f(x) = x³ − 5x² + 3x − 7 is divided by (x − 3)?
Q 2Polynomials
If f(x) = 2x³ + kx − 4 and (x − 2) is a factor, what is the value of k?
📐 Unit 2 — Quadratic Functions Q 3–4
📘 Key Concept
A quadratic f(x) = ax² + bx + c has vertex at x = −b/(2a). The discriminant b² − 4ac tells the nature of roots.
Quadratic Formula: x = (−b ± √(b²−4ac)) / 2a
Discriminant: Δ = b²−4ac
Δ > 0 → 2 real roots | Δ = 0 → 1 real | Δ < 0 → 2 complex
📝 Example
Vertex of f(x) = 2x² − 8x + 5:
x = 8/4 = 2; y = 2(4) − 16 + 5 = −3 → Vertex (2, −3) ✓
Q 3Quadratics
The discriminant of 3x² − 5x + 4 = 0 indicates the equation has:
Q 4Quadratics
Which vertex form represents f(x) = x² − 6x + 11?
🔮 Unit 3 — Complex Numbers Q 5–6
📘 Key Concept
The imaginary unit i = √(−1), so i² = −1, i³ = −i, i⁴ = 1 (cycle of 4). Complex numbers: a + bi.
(a+bi)(c+di) = (ac−bd) + (ad+bc)i
Conjugate of (a+bi) = (a−bi)
|a+bi| = √(a²+b²)
📝 Example
(3 + 2i)(1 − 4i) = 3 − 12i + 2i − 8i² = 3 − 10i + 8 = 11 − 10i
Answer: 11 − 10i ✓
Q 5Complex Numbers
Simplify (2 + 3i)(4 − i). What is the result?
Q 6Complex Numbers
What is i⁴³?
🔍 Unit 4 — Rational Root & Zeros Q 7–8
📘 Key Concept
The Rational Root Theorem: possible rational zeros of aₙxⁿ + … + a₀ are ±(factors of a₀)/(factors of aₙ). Descartes' Rule of Signs counts sign changes for positive/negative real roots.
Possible rational zeros = ±p/q
p = factors of constant term
q = factors of leading coefficient
📝 Example
f(x) = 2x³ − 3x + 1: possible rational roots = ±1, ±1/2
Test x=1: 2−3+1 = 0 ✓ → (x−1) is a factor
Q 7Rational Roots
According to the Rational Root Theorem, which is a possible rational zero of f(x) = 6x³ − 5x² + 3x − 10?
Q 8Fundamental Theorem
A degree-4 polynomial with real coefficients has zeros 2+i and −3. What is another guaranteed zero?
🌱 Unit 5 — Radicals & Rational Exponents Q 9–10
📘 Key Concept
Rational exponents: x^(m/n) = (ⁿ√x)ᵐ. Always check for extraneous solutions when solving radical equations.
a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ)
√(ab) = √a · √b
To solve √(f(x))=g(x): square both sides, then CHECK
📝 Example
Solve √(x+5) = 3 → x+5 = 9 → x = 4. Check: √9 = 3 ✓
Q 9Radical Equations
Solve: √(x + 2) = x − 4
Q 10Rational Exponents
Simplify: 27^(2/3)
📈 Unit 6 — Exponential & Logarithmic Functions Q 11–12
📘 Key Concept
log_b(x) = y ⟺ bʸ = x. The Change of Base formula converts any log. Key properties: product, quotient, power rules.
log_b(MN) = log_b(M) + log_b(N)
log_b(M/N) = log_b(M) − log_b(N)
log_b(Mᵖ) = p · log_b(M)
Change of Base: log_b(x) = ln(x)/ln(b)
📝 Example
Solve: log₂(x) = 5 → x = 2⁵ = 32
Q 11Logarithms
Solve for x: log₃(x − 2) + log₃(x + 4) = 3
Q 12Exponential Growth
If a population doubles every 6 years and starts at 500, approximately how many years until it reaches 8,000?
➗ Unit 7 — Rational Functions Q 13
📘 Key Concept
For f(x) = p(x)/q(x): Vertical asymptotes where q(x)=0 (if not cancelled). Horizontal asymptotes: compare degrees of p and q.
deg(p) < deg(q) → HA: y = 0
deg(p) = deg(q) → HA: y = leading coefficients ratio
deg(p) > deg(q) → No HA (oblique asymptote)
Q 13Rational Functions
What is the horizontal asymptote of f(x) = (3x² − 5) / (x² + 7)?
🔢 Unit 8 — Sequences & Series Q 14–15
📘 Key Concept
Arithmetic: aₙ = a₁ + (n−1)d, Sₙ = n/2(a₁+aₙ). Geometric: aₙ = a₁·rⁿ⁻¹, Sₙ = a₁(1−rⁿ)/(1−r). Infinite geometric series: S = a₁/(1−r) when |r| < 1.
Arithmetic nth term: aₙ = a₁ + (n−1)d
Geometric nth term: aₙ = a₁ · rⁿ⁻¹
Infinite Geo Sum: S∞ = a₁/(1−r), |r|<1
Q 14Sequences
Find the sum of the first 10 terms of the arithmetic sequence: 3, 7, 11, 15, …
Q 15Series
Find the sum of the infinite geometric series: 8 + 4 + 2 + 1 + …
⭕ Unit 9 — Conic Sections Q 16–17
📘 Key Concept
Standard forms: Circle (x−h)²+(y−k)²=r². Ellipse x²/a²+y²/b²=1. Hyperbola x²/a²−y²/b²=1. Parabola x²=4py (vertical) or y²=4px (horizontal).
Circle: (x−h)² + (y−k)² = r²
Ellipse: x²/a² + y²/b² = 1 (a > b > 0)
Parabola y = a(x−h)² + k: vertex (h,k)
Q 16Conic Sections
What is the center and radius of the circle: x² − 4x + y² + 6y = 12?
Q 17Conic Sections
The equation x²/25 + y²/9 = 1 represents an ellipse. What are the lengths of the major and minor axes?
🎲 Unit 10 — Statistics & Probability Q 18–20
📘 Key Concept
Binomial probability: P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ. Normal distribution: 68-95-99.7 rule. Standard deviation and variance measure spread.
Binomial: P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ
Permutation: P(n,r) = n!/(n−r)!
Combination: C(n,r) = n!/[r!(n−r)!]
Q 18Statistics
A fair coin is flipped 6 times. What is the probability of getting exactly 4 heads?
Q 19Combinations
How many ways can a committee of 4 be chosen from 9 people?
Q 20Binomial Theorem
Using the Binomial Theorem, find the coefficient of x³ in the expansion of (x + 2)⁵.
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