Factor Theorem

In this lesson, we will look at the Factor Theorem

When f(x) is divided by x – a, we get

f(x) = (x – a)Q(x) + remainder

From the Remainder Theorem, we get

f(x) = (x – a)Q(x) + f(a)

If f(a) = 0 then the remainder is 0 and

f(x) = (x – a)Q(x)

We can then say that x – a is a factor of f(x)

The Factor Theorem states that

x – a is a factor of the polynomial f(x) if f(a) = 0

Example:

Determine whether x + 1 is a factor of the following polynomials.

a) 3x⁴ + x³ – x² + 3x + 2

b) x⁶ + 2x(x – 1) – 4

Solution:

a) Let f(x) = 3x⁴ + x³ – x² + 3x + 2

f(–1) = 3(–1)4 + (–1)3 – (–1)2 +3(–1) + 2

= 3(1) + (–1) – 1 – 3 + 2 = 0

Therefore, x + 1 is a factor of f(x)

b) Let g(x) = x⁶ + 2x(x – 1) – 4

g(–1) = (–1)6 + 2(–1)( –2) –4 = 1

Therefore, x + 1 is not a factor of g(x)

Videos

The Factor Theorem and The Remainder Theorem
What the theorems are and how they can be used to find the linear factorization of a polynomial.

This video demonstrates how to use the Factor Theorem to factor polynomials.

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