**A REVIEW OF FACTORING
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

Definition of FactoringFactoring actually means to "divide out". When you are asked to "factor" a polynomial expression, you are actually told to write the expression as a product of factors. Often you are asked to factor "relative to the integers." This means that your factors cannot include rational, irrational, or imaginary numbers.

Some Factoring Methods

A. Factor out the Greatest Common Factor.

- Find the greatest common numeric factor of all of the coefficients.
- Find the greatest common factor of all of the variables.
- Divide out the greatest common factor
Example 1:

Given the polynomial expression

, factor out the3x^{7}18x^{3}+ 9x^{2 }Greatest Common Factor.We can easily find the greatest common numeric factor by "casual observation." It is

. The greatest common variable factor is3.x^{2}In summary,

3x^{7}18x^{ }+ 9x^{2 }.= 3x^{2}(x^{ 5 }6x + 3)Example 2:

Given the polynomial expression

8, factor out thex^{2}+ 4Greatest Common Factor.We can easily find the greatest common numeric factor by "casual observation." It is

. There is no greatest common variable factor.4In summary,

8x^{2}+ 4.= 4(2x^{2}+ 1)Example 3:

Given the polynomial expression

, factor out thex(x 3) (x 3)Greatest Common Factor.This expression has two terms and both contain the factor

. This is the Greatest Common Factor and we will factor it out as follows:x 3

2x(x 3) (x 3) = (x 3)( 2x 1)Please note that "factoring" means to "divide out". When we divide

out of the first term, we are left with a factor ofx 3; and when we divide2xout of the second term we are left with a factor ofx 3.1

B. Factor relative to the integers using the Grouping Method.

- Collect the terms in an expression into two groups.
- Factor out the greatest common factor from each group. You should end up with an identical multinomial factor in each term. If not, regroup, then factor out the greatest common factor again. If you still don't end up with an identical multinomial factor in each term, the expression is NOT factorable by grouping.
- Factor out the identical multinomial factor from each term.
Example 4:

Try to factor the polynomial expression

relative to the integers using thex^{3}4x^{2}+ 2x 8Grouping Method.Let's form two groups as follows:

(x^{3}4x^{2}) + (2x 8)Please note that these were the most convenient groups. It just so happens that when we factor common factors out of both, we have an identical binomial factor in each term!

xNote that we now have an identical binomial factor in each term!^{2 }(x 4) + 2(x 4)Lastly, we will factor out the identical binomial factor

as follows:(x 4)

(x 4)(x^{2 }+2)In summary,

x^{3}4x^{2}+ 2x 8 =(x 4)(x^{2}+.2)

C. Factoring the Trinomialaxrelative to the Integers.^{ 2}+ bx + c

- Find two integers whose product equals
and whose sum equalsac. These integers are always factors ofb. If this is not possible, the trinomial cannot be factored relative to the integers.ac- Replace the coefficient
of the middle termbwith the sum of the integers and distribute the variable.bx- Use the factoring by grouping method to find a product of two factors.
Example 5:

Try to factor the trinomial

2xrelative to the integers.^{2}+ 7x + 6Given the general trinomial

ax, we find that^{2}+ bx + c,a = 2, andb = 7, andc = 6.ac = 2(6) = 12We need to find two integers whose

productequalsand whose12sumequals. However, the guessing is simplified because the integers MUST be factors of7.12Let's write down the factor pairs of

:12

1, 122, 63, 41, 122, 63, 4It seems that the two integers we need are

and3since4and3(4) = 123.+ 4 = 7Now we write

2xas follows:^{2 }+ 7x + 6

2x^{2}+ 3x + 4x + 6Next, we'll use the factoring by grouping method to get

(2x.^{2 }+ 3x) + (4x + 6)Then, we factor the common factor out of both groups as follows:

x(2x + 3) + 2(2x + 3)Lastly, we will factor out the identical binomial factor to get

.(2x + 3)(x + 2)In summary,

2x^{2 }+ 7x + 6.= (2x + 3)(x + 2)As a beginner you might want to use

FOILto convince yourself that you have indeed found the correct factors.Example 6:

Try to factor the trinomial

6relative to the integers.x^{2}+ x 2Given the general trinomial

ax, we find that^{2 }+ bx + c,a = 6, andb = 1, andc = 2.ac = 6() = 122We need to find two integers whose

productequalsand whose12sumequals.1However, the guessing is simplified because the integers MUST be factors of

.12Let's write down the factor pairs of

:12

1, 122, 63, 41, 122, 63, 4It seems that the two integers we need are

and3since4and3(4) = 123.+ 4 = 1Now we write

6as follows:x^{2 }+ x 2

6x^{2 }3x + 4x 2Next, we'll use the factoring by grouping method to get

(6x.^{2 }3x) + (4x 2)Then, we factor the common factor out of both groups as follows:

3x(2x1) + 2(2x 1)Lastly, we will factor out the identical binomial factor to get

(2x.1)(3x + 2)In summary,

6x^{2}+ x 2.= (2x^{ }1)(3x + 2)As a beginner you might want to use

FOILto convince yourself that you have indeed found the correct factors.Example 7:

Try to factor the trinomial

xrelative to the integers.^{2}5x + 6Given the general trinomial

ax, we find that^{2}+ bx + c,a = 1, andb = 5, andc = 6.ac = 1(6) = 6We need to find two integers whose

productequalsand whose6sumequals.5However, the guessing is simplified because the integers MUST be factors of

.6Let's write down the factor pairs of

:6

6, 12, 32, 31, 6It seems that the two integers we need are

and2since3and2(3) = 62.+ (3) = 5Now we write

as follows:x^{2}5x + 6

x^{2 }2x 3x + 6Next, we'll use the factoring by grouping method to get

(x.^{2}2x) + (3x + 6)Then, we factor the common factor out of both groups as follows:

x(x2) 3(x 2)Lastly, we will factor out the identical binomial factor to get

(x.2)(x 3)In summary,

x^{2}5x + 6.= (x^{ }2)(x 3)As a beginner you might want to use

FOILto convince yourself that you have indeed found the correct factors.

D. Factoring "special" polynomials relative to the Integers.

Difference of Squares:Difference of Cubes:Sum of Cubes:

Please note that aSums of Squarescannot be factored relative to the integers. The factors ofSums of Squaresare imaginary! This will be discussed at a later time!Example 8:

Factor the following "special" polynomials relative to the integers.

(a)

x^{2}9Using the

Difference of Squaresformula withanda = 3a^{2}, we can factor as follows:= 9

(x3)(x+ 3)(b)

x^{3}8Using the

Difference of Cubesformula withanda = 2a^{2}, we can factor as follows:= 4

(x 2)(x^{2}+ 2x + 4)(c)

x^{3}+ 125Using the

Sum of Cubesformula,anda = 5a^{2}, we can factor as follows:= 25

(x + 5)(x^{2}5x + 25)