Chapter 8
SPI 3102.3.3: Factor Polynomials.
3102.3.9: Find two binomial factors of a quadratic expression.
Section 8.1
Objective: You will be able to find the Greatest Common Factor (GCF) of different terms in a polynomial.
How to find the GCF: Let’s start with just numbers no variables.

You need to make what is a called a factor tree so you can find the prime factorization of the number. To make
a factor tree you start making branches that have factors of the number given.
 You stop each branch once you have reached a number that is a prime number.
Prime number: A number that only has the factors of 1 and itself.
Examples:
I circle the
numbers once I
find ones that
are prime. So
that I know to
stop.
18
2
120
9
3
2
3
56
60
2
2
28
___
Prime factorization: 2*3*3
2
___
___ ___
PF: _____________
PF: ________________
So, to find the GCF of two or more numbers you need to list the prime factorization of each of the numbers and find what
they have in common. Let’s try it!!
Example 1: Find the GCF of 16 and 20.
1) make a factor tree for each number.
2) write out the prime factorization of
each number
3) find what they have in common, or
share.
4) multiply together and that will give
you the GCF.
16
20
2
8
2
2
4
2
2
5
2
PF: 2*2*2*2
Each number has 2 twos so the GCF is: 2*2 = 4
10
PF: 2*2*5
Example 2: Find the GCF of 100 and 120.
100
120
PF of 100: __________________
PF of 120: ___________________
What do they share? ________________
GCF: ____________
YOU TRY!!!
1) Find the GCF of 33 and 66.
2) Find the GCF of 84 and 42.
What to do if there are numbers and variables: You will do the same thing as above for the numbers and then you will need
find out how many of each variable is shared by each term.
Example 1: Find the GCF of 15x4 and 35x2.
* Make a factor tree for the numbers.
* List the prime factorization for each
Term only you need to add the variables
to it.
* Then decide what the share to find
the GCF.
Example 2: Find the GCF of 12p2 and 30q5.
x4
15
3
5
PF: 3*5*x*x*x*x
x2
35
5
7
PF: 5*7*x*x
They share a 5 and 2 x’s so the GCF is 5x2
Example 3: Find the GCF of 27y3z and 45x2y.
YOU TRY!!!
1) Find the GCF of 6t3 and 9t.
2) Find the GCF of 10gh2 and 5h.
Section 8.2
(SPI 3102.3.3)
Objective: You will use what you know about finding the GCF to factor out the GCF of a polynomial.
Factoring a GCF
- To factor out the GCF of a polynomial you have to find the GCF of all the terms.
- Once you know the GCF, then you will remove (divide) it from the terms in the
polynomial.
Example 1: Factor the following polynomial: 3n3 + 12n2
Step 1: List the prime factorization of each term.
3n3 = 3*n*n*n
12n2 = 2*2*3*n*n
Step 2: Find the GCF from the prime factorizations.
GCF: 3*n*n = 3n2
3𝑛3
Step 3: Divide out the GCF from each term.
3𝑛2
Step 4: Write the polynomial with the GCF in front of
the parentheses and what is left inside the parentheses.
Example 2:
Factor the following: 15x5 − 18x
=𝑛
12𝑛2
3𝑛2
=4
3n2 (n + 4)
Example 3:
Factor the following: – 6n6 + 18n4 − 24n
YOU TRY
1) Factor the following: 8c2 + 7c
2) Factor the following: −5m4 − 5m3 + 5m2
Factoring by grouping:
Sometimes the terms of a polynomial won’t have anything in common so you must factor by a
technique called grouping.
Example 1: Factor 8x3 + 6x2 + 20x + 15.
Step 1: Group terms that have common factors.
(8x3 + 6x2) + (20x + 15)
Step 2: Identify and factor the GCF out of each group.
(8x3 + 6x2) + (20x + 15)
GCF on 2nd: 5
GCF of 1st: 2x2
**Put the common factor in front and what’s left
in the parentheses.
2x2(4x + 3) + 5(4x + 3)
2x2(4x + 3) + 5(4x + 3)
Step 3: Factor out the common binomial factor.
They both share the term (4x + 3)
**Put the common factor in front and what’s left
in the parentheses.
Example 2: Factor: 21x3 + 12x2 + 14x + 8
(4x + 3)(2x2 + 5)
Example 3: Factor: 40x3 - 50x2 + 12x – 15
Group: (21x3 + 12x2) + (14x + 8)
YOU TRY!!!!
1) Factor: 12n3 − 15n2 − 8n + 10
2) Factor: 2x3 + 8x2 + 3x + 12
Sections 8.3
Objective: You will be able to factor quadratics of the form Ax2 + Bx + C when A = 1.
Factoring in the form x2 + Bx + C:
We will start with the simplest form of a quadratic trinomial. This is when the leading coefficient,
A, is equal to 1.
Example: x2 + 2x + 1
But first, remember the process
of multiplying two binomials?
(x + 2)(x +3)
x(x) + x(3) + 2(x) + 2(3)
x2 + 3x +2x + 6
x2 + 5x + 6
Now we must go backwards!
Given x2 + 5x + 6, can you find its factors?
Example 1: Factor x2 + 5x + 6
Step 1: Create your parenthesis and split up your first term.
(x
Step 2: Find the factors of your third term.
_6_
1, 6
2, 3
Step 3: Choose the factors that add to get your middle term.
Step 4: Complete your parentheses with your chosen factors.
)(x
(x + 2)(x + 3)
*You can check your answer by multiplying the binomials together.
Example 2:
Factor d2 + 7d − 8
Example 3:
Factor m2 – 6m – 27
YOU TRY:
1) Factor c2 + 11c + 10
2) Factor x2 – x – 12
3) Factor x2 + 8x – 20
4) Factor a2 + 16a + 64
)
2+3=5
Sections 8.4
Objective: You will be able to factor quadratics of the form Ax2 + Bx + C when A > 1.
Factoring in the form Ax2 + Bx + C:
Now, we need to know how to factor a quadratic trinomial when the leading coefficient, A > 1.
Example: 3x2 + 5x + 2
Example 1: 2x2 + 7x + 5
Step 1: Identify your coefficients A, B, and C.
A: 2
B: 7
Step 2: Multiply A and C.
2 ∙ 5 = 10
1
2
Step 3: Find 2 factors of AC that add to get B.
Step 4: Rewrite your middle term using the factors.
C: 5
10
5
2x2 + 2x + 5x + 5
*Does it matter which order
to write the middle term? _________
Step 5: Use Factor by Grouping.
2x(x + 1) + 5(x + 1)
(x + 1)(2x + 5)
Example 2:
Factor 3x2 – 16x + 5
Example 3:
Factor 5y2 + 8y – 4
You Try!
1. 7m2 + 6m – 1
2. 2r2 + 7r – 30
Section 8.5 and 8.6
Special Case Factoring
Perfect Squares
Factor the following polynomials.
1. x2 + 4x + 4
2. x2 + 6x + 9
3. x2 + 10x + 25
4. x2 + 18x + 81
What do you notice about the two factors in each problem?
How is the constant in the factor related to the middle coefficient?
How is the constant in the factor related to the 3rd term?
Using this pattern, factor the following polynomials.
1. x2 + 16x + 64
2. 81x2 + 90x + 25
3. 9x2 – 15x + 64
4. 4x2 – 12x + 9
5. a2 + 20a + 100
6. 4x2 – 28x + 49
Difference of Two Squares
Multiply the following binomials.
1. (x + 4)(x – 4)
2. (x + 6)(x – 6)
3. (x + 12)( x – 12)
3. (x + 7)(x – 7)
What do you notice about the two binomial factors before you multiplied?
What happens with the middle term after you simplify the multiplication?
Using this pattern, factor the following polynomials.
1. x2 – 81
2. k2 – 25
3. 4y2 – 361
4. 9m2 – 169
5. 100x2 – 4y2
6. 3p2 – 9q4
Checking for a GCF
Sometimes, we can factor out a GCF from our problem to make the factoring easier.
Example 1: Factor 3x2 + 21x + 18
Step 1: Find the GCF.
Step 2: Factor out the GCF.
3∙𝑥∙𝑥
3𝑥 2
3
+
21𝑥
3
3∙7∙𝑥
+
18
3
2∙3∙3
𝐺𝐶𝐹 = 3
= 𝑥 2 + 7𝑥 + 6
Step 3: Factor the polynomial that is left.
𝑥 2 + 7𝑥 + 6 = (𝑥 + 1)(𝑥 + 6)
Step 4: Write all of your factors together.
3(𝑥 + 1)(𝑥 + 6)
Example 2:
Factor 8x6y2 – 18x2y2
Example 3:
Factor –8x3 + 12x2 + 20x
You Try!
1. 10k2 + 40k – 50
2. 2x2y – 2y3
3. –x2 – 4x + 60
4. –21mn2 + 112mn – 140m