2.1. Factoring Review http://www.ck12.org
Guidance
A polynomial is a sum of a finite number of terms. Each term consists of a constant that multiplies a variable. The
variable may only be raised to a non-negative exponent. The lettersa,b,c...in the following general polynomial
expression stand for regular numbers like 0, 5 ,−^14 ,√2 and thexrepresents the variable.
axn+bxn−^1 +...+f x^2 +gx+h
You have already learned many properties of polynomials. For example, you know the commutative property which
states that terms of a polynomial can be rearranged to create an equivalent polynomial. When two polynomials
are added, subtracted or multiplied the result is always a polynomial. This means polynomials are closed under
addition, and is one of the properties that makes the factoring of polynomials possible. Polynomials are not closed
under division because dividing two polynomials could result in a variable in the denominator, which is a rational
expression (not a polynomial).
There are three methods for factoring that are essential to master. The first method you should always try is to
factor out the greatest common factor (GCF) of the expression (see Example A). The second method you should
implement after factoring out a GCF is to see if you can factor the expression into the product of two binomials (see
Example B). This type of factoring is usually recognizable as a trinomial wherex^2 has a coefficient of 1. The third
type of basic factoring is the difference of squares. It is recognizable as one square monomial being subtracted from
another square monomial.
The rigor of the following factoring examples and exercises is greater than an introductory level factoring lesson but
the techniques are the same.
Example A
Use the GCF technique to factor the following expression. Check that the factored expression matches the original.
−^12 x^4 +^72 x^2 − 6
Solution:Many students just learning factoring may conclude that the three terms share no factors besides one. How-
ever, the name GCF is deceiving because this expression has an infinite number of equivalent expressions many of
which are more useful. In order to find these alternative expressions you must strategically factor numbers that are
neither the greatest factor nor common to all three terms. In this case,−^12 is an excellent choice.
−^12 x^4 +^72 x^2 − 6 =−^12 (x^4 − 7 x^2 + 12 )
In order to check to see that this is an equivalent expression, you need to distribute the−^12. When you distribute, the
first coefficient matches because it just gets multiplied by 1, the second term becomes^72 and the third term becomes
-6.
Example B
Factor the expression from Example A into the product of two binomials and a constant.
−^12 (x^4 − 7 x^2 + 12 )
Solution: Many students familiar with basic factoring may be initially stuck on a problem like this. However, you
should recognize that beneath the 4thdegree and the−^12 the problem boils down to being able to factoru^2 − 7 u+
12 which is just(u− 3 )(u− 4 ).
Start by rewriting the problem:−^12 (x^4 − 7 x^2 + 12 )
Then choose a temporary substitution:Letu=x^2.
Then substitute and factor away. Remember to substitute back at the end.