Factors of Polynomials

From: Barbara Hinton
Date: 28 February 1999
Subject: Factors of Polynomials

How can I show that (x+2) is a factor of

f(x) = x^4 + 2x^3 - 7x^2 - 8x + 12

without doing long division?

You are right - there is a 'short cut' method:

If a is a root of a polynomial, then
(x - a) is a factor of that polynomial

All you have to do, to show that (x + 2) is a factor of f(x), is to evaluate f(-2).

In other words, work out f(x) with x = -2:

f(-2) = (-2)⁴ + 2×(-2)³ - 7×(-2)² - 8×(-2) + 12

f(-2) = 16 - 16 - 28 + 16 + 12

f(-2) = 0

Because f(-2) = 0, (x + 2) is a factor of f(x).