Lesson 5-4: Factoring Polynomials
After this lesson, TSWBAT

factor polynomials completely (SOL A2.1d)
Factors
When you were younger, you learned that the factors of a number were numbers that divided
another without a remainder. In other words the numbers that you multiply together to make
another number are factors of that product. Take for example 18, its factors are 1, 2, 3, 6, 9 and 18.
It is sometimes easier to think of those factors in pairs as shown below:
Let’s factor 20 together.
Your Turn!
Find the factors of the following numbers. Factor them as shown above.
25
32
100
120
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Factors of a Polynomial
Just like with numbers the factors of a polynomial are two or more polynomials that you
multiply together to get that polynomial. Take a look at the examples below.
Greatest Common Factor
In the first example of factors of a polynomial we used the distributive property to multiply
3c(4c – 11) to get the product of 12c2 - 33c. We can use the distributive property to factor
polynomials. To do so we need to find the Greatest Common Factor, or GCF, of the terms
of the polynomial. The GCF is the largest common factor of a number. At this point in your math
career, you probably don't remember how to find the GCF, you might just know what it is, to
refresh your memory take a look at the example below:
Example 1
Find the GCF of 18 and 45.
2
When finding the GCF of monomials containing variables, you do the same for the coefficients.
However you need to also find the GCF of the variables. It always turns out to be the least power
of each variable. The next example illustrates this.
Example 2
Find the GCF of 16x5, 20x2 and 12x3.
Your Turn!
Match the following sets of monomials to the GCFs.
______1.
a.
______2.
b.
______3.
c.
______4.
d.
______5.
e.
______6.
f.
Factoring Using the Distributive Property
We can use the distributive property and the GCF to factor some polynomials and in fact, a
greatest common factor is always the first thing we should look for when factoring a polynomial.
Take a look at the following examples:
Example 3
Example 4
Factor: v2 + 4v
Factor: 10x3 - 25x2 + 20x
We say that the expression is factored or written in factored form.
3
Your Turn!
Factor the following polynomials.
6x - 4
2t2 - 10t4
9m12 - 36m7 + 81m5
24x3 - 96x2 + 48x
Factor By Grouping
We can also use the distributive property to factor polynomials with 4 terms, we call this factor
by grouping. The following examples illustrate how to actually do factoring by grouping.
Example 5
Example 6
Factor: 4n3 + 8n2 – 5n – 10
Factor: 5t4 + 20t2 + 6t + 24
Sometimes we are able to factor out a GCF right off the bat and we should always look for that
first. The next example shows this.
Example 7
Factor: 12p4 + 10p3 – 36p2 – 30p
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Your Turn!
Factor the following polynomial. Then, choose it’s factors from the bank at the right.
12v3 - 32v2 + 6v - 16
Factor Bank
(2v2 + 1)
(3v + 8)
(2v2 - 1)
(3v - 8)
2
(4v2 + 2)
Factoring Trinomials
While there are several methods for factoring trinomials, if a trinomial is factorable it can always
be done using factor by grouping. We are going to look at two cases, when the trinomial is in the
form x2 + bx + c and ax2 + bx + c.
Factoring x2 + bx + c
We can use patterns or factor by grouping to factor trinomials like x2 + 7x + 12 or d2 - 17d + 42.
We will first look at factor by grouping, then we will look at the patterns involved. Take a look at
these examples.
Example 8
Example 9
Factor x2 + 7x + 12
Factor d2 - 17d + 42
Can we do this an easier way?
Can we do this an easier way?
YES/NO?
YES/NO?
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Your Turn!
Factor the following polynomials, the easy way!
m2 - 9m + 8
y2 + y - 20
k2 + 10k + 16
t2 - 10t – 75
More Factoring Trinomials
When trinomials have a coefficient in front of the squared term like, 2y2 + 5y + 2 or 7x2 - 26x - 8,
they can also be factored using factor by grouping. The only difference is instead of finding
factors of the third term, you must find factors of the product of the coefficient in front of the
squared term and the third term.
Example 10
Example 11
Factor 2y2 + 5y + 2
Factor 7x2 - 26x - 8
Can we do this an easier way?
Can we do this an easier way?
YES/NO?
YES/NO?
Your Turn!
Factor the following trinomials.
3x2 - 17x + 10
8y2 - 10y – 3
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Factoring Binomials
You have looked at factoring polynomials using the distributive property and by using factor by
grouping. Now you will learn how to factor special binomials called difference of squares. A
difference of squares is a special binomial. You can recognize it by first looking for subtraction,
then perfect squares. Recall that perfect squares are numbers that have whole number square
roots. A difference of squares can be factored as follows
Difference of
Squares
Look at me!
I’m
important!
If the binomial has addition or if either term is not a perfect square then the binomial can't be
factored in this manner. Of course you should look to see if there is a GCF between the terms.
Below are examples of factoring difference of squares.
Example 12
Example 13
Example 14
Factor x2 - 49
Factor 25a2 - 81
Factor 16x2 + 25
Your turn!
Factor the following binomials.
4x2 - 100
49y2 - 4
9c2 - 64d2
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Sum and Difference of Cubes
All of the factoring we have done so far you learned in Algebra I. New to you should be the sum
and difference of cubes. There is a definite pattern to follow when factoring sum and
difference of cubes. First though you need to recognize the sum and difference of cubes. They are:


binomials (two terms)
both terms are perfect cubes
Remember a perfect cube is a number generated by multiplying another by itself 3 times. Some
examples of perfect cubes are:



8, 2(2)(2) = 8
64, 4(4)(4) = 64
1000, 10(10)(10)
You should be able to recognize these numbers. Here are the first few:
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Examples of sum of cubes are:


x3 + 27
8y3 +1000
Examples of difference of cubes are:


x3 - 64
27y3 -512
To factor the sum and difference of cubes use the following patterns:
Sum of Cubes
Difference of Cubes
Take a look at the following examples:
Example 15
Example 16
Factor g3 - 125
Factor 16x2 + 54
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Practice
9
Simplifying Quotients

In the last lesson you learned how to simplify the quotient of two polynomials by using long division or
synthetic division. Some quotients can be simplified by using factoring.
Example 17
Example 18
Simplify.
Simplify.
Practice
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