Factor Theorem

Lesson Objectives

State and explain the Factor Theorem.
Use the Factor Theorem to determine if a given expression is a factor of a polynomial.
Apply the Factor Theorem to solve polynomial equations.
Factorize polynomials completely using the Factor Theorem.
Solve real-world problems involving polynomial factorization.

Lesson Introduction

Imagine you are designing a roller coaster and need to calculate safe paths. You’ll work with polynomial curves to ensure they pass through required points. The Factor Theorem is a powerful tool to help test whether a certain path (or value) will fit into your curve equation. This lesson will bridge algebra with real-life applications like engineering and physics.

Core Lesson Content

The Factor Theorem states: If $f(c) = 0$ , then $(x - c)$ is a factor of the polynomial $f(x)$ .
Conversely, if $(x - c)$ is a factor of $f(x)$ , then $f(c) = 0$ .

Worked Examples

Example 1 (Basic):
Given

f(x) = x^2 - 5x + 6

, show that

(x - 2)

is a factor.

f(2) = (2)^2 - 5(2) + 6

= 4 - 10 + 6

= 0

\Rightarrow (x - 2) \text{ is a factor}

Example 2 (Intermediate):
Is

(x + 1)

a factor of

f(x) = x^3 + x^2 - x - 1

f(-1) = (-1)^3 + (-1)^2 - (-1) - 1

= -1 + 1 + 1 - 1

= 0

\Rightarrow (x + 1) \text{ is a factor}

Example 3 (Challenging):
Factor

f(x) = x^3 - 4x^2 + x + 6

completely using the Factor Theorem.

f(2) = 8 - 16 + 2 + 6 = 0

\Rightarrow (x - 2) \text{ is a factor}

f(x) = (x - 2)(x^2 - 2x - 3)

= (x - 2)(x - 3)(x + 1)

Example 4 (WAEC-style):
Factor

x^3 + 3x^2 - 4

completely.

f(1) = 1 + 3 - 4 = 0

\Rightarrow (x - 1) \text{ is a factor}

f(x) = (x - 1)(x^2 + 4x + 4)

= (x - 1)(x + 2)^2

Example 5 (Advanced):
Solve

x^3 + x^2 - 4x - 4 = 0

using the Factor Theorem.

f(2) = 8 + 4 - 8 - 4 = 0

\Rightarrow (x - 2) \text{ is a factor}

f(x) = (x - 2)(x^2 + 3x + 2)

= (x - 2)(x + 1)(x + 2)

Exercises

$\text{Use the Factor Theorem to test if } (x + 3) \text{ is a factor of } x^3 + 3x^2 - x - 3.$
$\text{Factorize } x^3 - x^2 - 4x + 4 \text{ completely.}$
$\text{If } (x - 2) \text{ is a factor of } f(x) = x^3 + ax^2 + bx + c, \text{ and } f(2) = 0, \text{ find a relationship between } a, b, \text{ and } c.$
$\text{Factor } f(x) = x^3 + 2x^2 - x - 2 \text{ completely.}$
[WAEC] $\text{Show that } (x + 2) \text{ is a factor of } x^3 + x^2 - 4x - 4.$ (Past Question)
[NABTEC] $\text{Factorize } x^3 - 7x + 6 \text{ completely.}$ (Past Question)
[JAMB] $\text{Determine whether } (x - 1) \text{ is a factor of } x^3 - 3x^2 + 3x - 1.$ (Past Question)
$\text{Factor } x^3 + 4x^2 + 3x.$
$\text{Given } f(x) = x^3 - 2x^2 - x + 2, \text{ find all factors.}$
[WAEC] $\text{Solve } x^3 - 3x + 2 = 0 \text{ using Factor Theorem.}$ (Past Question)

Conclusion / Recap

In this lesson, you’ve learned how to identify and use the Factor Theorem to determine factors of polynomials and solve equations. You also practiced complete factorization using this technique.
Next up: Synthetic Division and Remainder Theorem — a shortcut to polynomial division!