Unit 3 · 11th Grade Mathematics
Polynomials.
Features, operations, identities, and the factor/remainder theorems — all in one guide.
Topic A: Features of Polynomial Functions
Lessons 1–5 · Degree, end behavior, zeros, and graphs.
01–02 — Lessons 1–2
Degree, Leading Term & End Behavior
The degree and leading coefficient determine how a polynomial behaves at the far left and right.
- Degree: highest exponent in the polynomial.
- Leading coefficient: coefficient of the highest-degree term.
- Even degree, + leading: both ends go up (↑↑).
- Even degree, − leading: both ends go down (↓↓).
- Odd degree, + leading: left down, right up (↓↑).
- Odd degree, − leading: left up, right down (↑↓).
f(x) = −2x³ + 5x → degree 3 (odd), leading −2 (negative) → ↑↓ end behavior.
Practice — Lessons 1–2
Q1 · Multiple Choice
For f(x) = 4x⁴ − 3x² + 1, describe end behavior.
Q2 · Short Answer
What is the degree of p(x) = 7x⁵ − 3x³ + x − 2?
03–05 — Lessons 3–5
Zeros, Multiplicity & Graphs
The zeros of a polynomial correspond to x-intercepts. Multiplicity affects how the graph behaves at each zero.
- Zero of multiplicity 1: graph crosses the x-axis.
- Zero of multiplicity 2 (even): graph touches and bounces off x-axis.
- Zero of multiplicity 3 (odd > 1): graph crosses but flattens.
- Max # of zeros = degree of the polynomial.
- Use factored form to identify zeros: f(x) = a(x−r₁)(x−r₂)…
f(x) = (x−2)²(x+1) → zero at x=2 (mult. 2, bounce), x=−1 (mult. 1, cross).
Practice — Lessons 3–5
Q3 · Multiple Choice
At a zero of even multiplicity, the graph of a polynomial...
Q4 · Multiple Choice
How many zeros (real or complex) does a degree-5 polynomial have?
Topic B: Operating with Polynomials
Lessons 6–9 · Addition, multiplication, and division of polynomials.
06–07 — Lessons 6–7
Adding, Subtracting & Multiplying Polynomials
Polynomials are closed under addition, subtraction, and multiplication — the result is always a polynomial.
- Add/Subtract: combine like terms (same degree).
- Multiply: use the distributive property (FOIL for two binomials, then for larger).
- Degree of product = sum of degrees of the two polynomials.
(x+2)(x²−x+3) = x³−x²+3x + 2x²−2x+6 = x³ + x² + x + 6
Practice — Lessons 6–7
Q5 · Short Answer
Expand (x+3)(x−5). Give the constant term.
Q6 · Multiple Choice
What is the degree of the product of a degree-2 and a degree-3 polynomial?
08–09 — Lessons 8–9
Polynomial Long Division
Divide a polynomial by another using a process similar to numeric long division.
- Dividend ÷ Divisor = Quotient + Remainder/Divisor
- Step 1: Divide the leading term of the dividend by the leading term of the divisor.
- Step 2: Multiply, subtract, bring down.
- Step 3: Repeat until the remainder has lower degree than the divisor.
- Synthetic division: shortcut when dividing by (x − c).
(x³ − 2x + 4) ÷ (x − 1) → quotient x² + x − 1, remainder 3.
Practice — Lessons 8–9
Q7 · Multiple Choice
When can you use synthetic division instead of polynomial long division?
Q8 · Multiple Choice
If (x³ + 5x − 6) ÷ (x − 1) has remainder 0, what can you conclude?
Topic C: Polynomial Theorems & Identities
Lessons 10–14 · Remainder theorem, factor theorem, and polynomial identities.
10–12 — Lessons 10–12
Remainder & Factor Theorems
- Remainder Theorem: when p(x) is divided by (x − c), the remainder = p(c).
- Factor Theorem: (x − c) is a factor of p(x) if and only if p(c) = 0.
- Use to test potential zeros quickly without full division.
- Rational Root Theorem: possible rational zeros are ±(factors of constant) / (factors of leading coefficient).
p(x) = x³ − 6x + 4. Test x = 2: p(2) = 8 − 12 + 4 = 0 → (x−2) is a factor.
Practice — Lessons 10–12
Q9 · Multiple Choice
By the Remainder Theorem, what is the remainder when p(x) = x³ + 2x − 5 is divided by (x − 1)?
Q10 · Multiple Choice
If p(3) = 0, what can you conclude by the Factor Theorem?
13–14 — Lessons 13–14
Polynomial Identities
Polynomial identities are equations that are true for all values of the variable.
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b² (difference of squares)
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- Use identities to factor, simplify, and solve problems efficiently.
Factor: x² − 25 = (x+5)(x−5) using difference of squares identity.
Practice — Lessons 13–14
Q11 · Multiple Choice
Which identity factored x² − 49?
Q12 · Short Answer
Expand (x + 4)² using the perfect square identity. Give the middle term coefficient.