EDGE Math 8

Factoring Polynomials

The idea of factoring for natural numbers can be extended to polynomials. Recall that a polynomial is a finite sum of monomials. Expressing a polynomial as a product is called factoring.

Consider the polynomial $6 a x + 9 a y .$ As you will learn in the next sections, this polynomial can be factored in any of the following ways:

6 a x + 9 a y = 1 (6 a x + 9 a y)

6 a x + 9 a y = - 1 (- 6 a x - 9 a y)

6 a x + 9 a y = a (6 x + 9 y)

6 a x + 9 a y = 3 (2 a x + 3 a y)

6 a x + 9 a y = 3 a (2 x + 3 y)

As you can see, there are at least five ways to write $6 a x + 9 a y$ as a product where each factor is a polynomial with integer coefficients.

Now consider the polynomial $2 x + 3 y .$ If you will write it as a product under the condition that each factor is a polynomial with integer coefficients, you will find that it can be written as a product in only two ways, namely

2 x + 3 y = 1 (2 x + 3 y);

and

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

Polynomials like $2 x + 3 y$ are called prime polynomials. Below are other examples of prime polynomials.

4 a c - 5 b

x^{2} + 4

y^{2} - 2 y + 2