3.2 Factoring Polynomials

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1. Quick Recap — What Is a Polynomial?

A polynomial is a mathematical expression made up of terms involving a variable (usually x), where each term has a whole-number power. For example:

• $x^3 - 3x^2 - 13x + 15$ is a cubic polynomial (highest power is 3)
• $x^2 - 2x - 15$ is a quadratic polynomial (highest power is 2)
• $x - 1$ is a linear polynomial (highest power is 1)

When we factorise a polynomial, we rewrite it as a product (multiplication) of simpler expressions — just like how $12 = 3 \times 4$.

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2. The Factor Theorem

The Factor Theorem is the most important tool for finding factors of a polynomial. Here is what it says:

In symbols:

$$\text{If } P(c) = 0, \text{ then } (x - c) \text{ is a factor of } P(x)$$

Think of it this way: A "factor" divides into the polynomial perfectly, leaving no remainder. The Factor Theorem gives you a shortcut — instead of doing a full division, you just substitute a value and check if you get zero.

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2.1 Using the Factor Theorem — Step by Step

Example: Show that $(x - 3)$ is a factor of $f(x) = x^3 - 6x^2 + 11x - 6$.

Step 1: Identify the value of $c$. Since the factor is $(x - 3)$, we have $c = 3$.

Step 2: Substitute $x = 3$ into the polynomial.

$f(3) = (3)^3 - 6(3)^2 + 11(3) - 6$ $= 27 - 54 + 33 - 6$ $= 0 ✓$

Step 3: Since $f(3) = 0$, we conclude that $(x - 3)$ is a factor of $f(x)$.

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2.2 The Extended Factor Theorem

Sometimes a factor looks like $(ax - b)$, where the number in front of $x$ is not 1 — for example, $(2x - 1)$.

In this case:

$$\text{If } P\!\left(\frac{b}{a}\right) = 0, \text{ then } (ax - b) \text{ is a factor of } P(x)$$

Why? Set $(ax - b) = 0$, which gives $x = \dfrac{b}{a}$. Substitute this value in.

Example: Check if $(2x - 1)$ is a factor of $f(x) = 2x^3 - x^2 - 2x + 1$.

Set $x = \dfrac{1}{2}$:

$f\!\left(\tfrac{1}{2}\right) = 2\!\left(\tfrac{1}{8}\right) - \left(\tfrac{1}{4}\right) - 2\!\left(\tfrac{1}{2}\right) + 1 = \tfrac{1}{4} - \tfrac{1}{4} - 1 + 1 = 0 ✓$

Since the result is zero, $(2x - 1)$ is a factor.