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Factoring Polynomials

Factoring a polynomial is the opposite process of multiplying polynomials. Recall that when we factor a number, we are looking for prime factors that multiply together to give the number; for example

6 = 2 ´ 3 , or 12 = 2 ´ 2 ´ 3.

When we factor a polynomial, we are looking for simpler polynomials that can be multiplied together to give us the polynomial that we started with. You might want to review multiplying polynomials if you are not completely clear on how that works.

§ When we factor a polynomial, we are usually only interested in breaking it down into polynomials that have integer coefficients and constants.

Simplest Case: Removing Common Factors

The simplest type of factoring is when there is a factor common to every term. In that case, you can factor out that common factor. What you are doing is using the distributive law in reverse—you are sort of un-distributing the factor.

Recall that the distributive law says

a(b + c) = ab + ac.

Thinking about it in reverse means that if you see ab + ac, you can write it as a(b + c).

Example: 2x² + 4x

Notice that each term has a factor of 2x, so we can rewrite it as:

2x² + 4x = 2x(x + 2)

Difference of Two Squares

If you see something of the form a² - b², you should remember the formula

Example: x² – 4 = (x – 2)(x + 2)

• This only holds for a difference of two squares. There is no way to factor a sum of two squares such as a² + b² into factors with real numbers.