# 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”.

__Example__:

Factor each polynomial by removing the common factor.

**(a) **$$20{x^2}y + 8xy$$

**(b) **$$u\left( {v + w} \right) + 7v\left( {v + w} \right)$$

__Solution__:

**(a) **Here $$4xy$$ is a common factor of the two terms, since

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

Therefore,

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

**(b) **Here the common factor is $$v + w$$, and we have

$$u\left( {v + w} \right) + 7v\left( {v + w} \right) = \left( {u + 7v} \right)\left( {v + w} \right)$$