Polynomial Functions
Grade 11 · Math · Study Guide
Explore polynomial functions, their graphs, zeros, and algebraic properties including the Factor and Remainder Theorems.
A polynomial function has the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₁x + a₀, where n is a non-negative integer and the leading coefficient aₙ ≠ 0. The degree n determines the function's general shape and the maximum number of real zeros.
End behavior is governed by the degree and the sign of the leading coefficient. Even-degree polynomials have both ends pointing the same direction; odd-degree polynomials have ends pointing in opposite directions.
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 dividing f(x) by (x − c) gives a remainder of f(c).
| Degree | Leading Coefficient | Left end | Right end |
|---|---|---|---|
| Even | Positive | Up (→ +∞) | Up (→ +∞) |
| Even | Negative | Down (→ −∞) | Down (→ −∞) |
| Odd | Positive | Down (→ −∞) | Up (→ +∞) |
| Odd | Negative | Up (→ +∞) | Down (→ −∞) |
| Theorem | Statement |
|---|---|
| Factor Theorem | f(c) = 0 ⟺ (x − c) is a factor of f(x) |
| Remainder Theorem | f(x) ÷ (x − c) leaves remainder f(c) |
| Fundamental Theorem of Algebra | A degree-n polynomial has exactly n complex zeros (counting multiplicity) |
Key terms
- Degree
- The highest exponent in a polynomial; it determines end behavior and the maximum number of zeros.
- Leading coefficient
- The coefficient of the highest-degree term; controls whether the graph rises or falls at the ends.
- Zero (root)
- A value c where f(c) = 0; the x-intercept of the polynomial's graph.
- Multiplicity
- The number of times a factor (x − c) appears; even multiplicity means the graph touches but does not cross the x-axis.
Practice questions
What is the degree of f(x) = 4x⁵ − 3x² + 7?
- 2
- 4
- 5
- 7
Show answer
The degree is the highest exponent, which is 5.
For f(x) = −2x⁴ + x − 1, what is the end behavior?
- Both ends up
- Both ends down
- Left up, right down
- Left down, right up
Show answer
Even degree with negative leading coefficient → both ends go to −∞.
Use the Factor Theorem to determine if (x − 3) is a factor of f(x) = x³ − 5x² + 6x.
Show answer
f(3) = 3³ − 5(3²) + 6(3) = 27 − 45 + 18 = 0, so (x − 3) is a factor.
If f(x) = (x − 2)²(x + 5), at x = 2 the graph will:
- Cross the x-axis
- Touch and turn at the x-axis
- Have a vertical asymptote
- Be undefined
Show answer
(x − 2) has multiplicity 2 (even), so the graph touches but does not cross at x = 2.
Find all real zeros of f(x) = x³ − x² − 6x.
Show answer
Factor: x(x² − x − 6) = x(x − 3)(x + 2). Zeros: x = 0, 3, −2.
Fun facts
- The Fundamental Theorem of Algebra guarantees every polynomial of degree n has exactly n zeros in the complex numbers — real zeros are just the special case.
- Roller coaster designers use polynomial curves called splines to create smooth, continuous track shapes without sudden jolts.
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