Factoring Polynomials

When two or more algebraic expressions are multiplied, each expression is called a factor of the product. For instance, in the product $$\left( {x – y} \right)\left( {x + y} \right)\left( {2{x^2} – y} \right)x$$, the factors are the $$x – y$$, $$x + y$$, $$2{x^2} – y$$ and $$x$$. Often we are given a product in its expanded form and we need to find the original factors. The process of finding these factors is called factoring.

In this tutorial, we confine our study of factoring to polynomials with integer coefficients. Thus, we shall not yet consider such possibilities as $$5{x^2} – {y^2} = \left( {\sqrt 5 x – y} \right)\left( {\sqrt 5 x + y} \right)$$, because $$\sqrt 5 $$ is not an integer.

Of course, we can factor any polynomial “trivially” by writing it as $$1$$ times itself or as $$ – 1$$ times its negative. A polynomial with integer coefficients that cannot be factored (except trivially) into two or more polynomials with integer coefficients is said to be prime. When a polynomial is written as a product of prime factors, we say that it is factored completely.


Removing a Common Factor

The distributive property can be used to factor a polynomial in which all the terms contain a common factor. The following example illustrates how to “remove the common factor”.



Factor each polynomial by removing the common factor.
(a) $$20{x^2}y + 8xy$$
(b) $$u\left( {v + w} \right) + 7v\left( {v + w} \right)$$


(a) Here $$4xy$$ is a common factor of the two terms, since
$$20{x^2}y = \left( {4xy} \right)\left( {5x} \right)$$   and   $$8xy = \left( {4xy} \right)\left( 2 \right)$$

$$20{x^2}y + 8xy = \left( {4xy} \right)\left( {5x} \right) + \left( {4xy} \right)\left( 2 \right) = 4xy\left( {5x + 2} \right)$$

(b) Here the common factor is $$v + w$$, and we have
$$u\left( {v + w} \right) + 7v\left( {v + w} \right) = \left( {u + 7v} \right)\left( {v + w} \right)$$