Chapter 13
Factoring
Polynomials

Martin-Gay,
Developmental Mathematics
2
13.1 – The Greatest Common Factor
13.2 – Factoring Trinomials of the Form
x
2 + bx + c
13.3 – Factoring Trinomials of the Form
ax
2 + bx + c
13.4 – Factoring Trinomials of the Form
x
2 + bx + c
by Grouping
13.5 – Factoring Perfect Square Trinomials and Diff
erence of Two Squares
13.6 – Solving Quadratic Equations by Factoring
13.7 – Quadratic Equations and Problem Solving
Chapter Sections

§ 13.1
The Greatest Common
Factor

Martin-Gay,
Developmental Mathematics
4
Factors
Factors
(either numbers or polynomials)
When an integer is written as a product of
integers, each of the integers in the product is a
factor
of the original number.
When a polynomial is written as a product of
polynomials, each of the polynomials in the
product is a
factor
of the original polynomial.
Factoring
– writing a polynomial as a product of
polynomials.

Martin-Gay,
Developmental Mathematics
5
Greatest common factor
– largest quantity that is a
factor of all the integers or polynomials involved.
Finding the GCF of a List of Integers or Terms
1)
Prime factor the numbers.
2)
Identify common prime factors.
3)
Take the product of all common prime factors.
•
If there are no common prime factors, GCF is 1.
Greatest Common Factor

Martin-Gay,
Developmental Mathematics
6
Find the GCF of each list of numbers.
1)
12 and 8
12 =
2
·
2
·
3
8 =
2
·
2
·
2
So the GCF is
2
·
2
= 4.
2)
7 and 20
7 = 1
·
7
20 = 2
·
2
·
5
There are no common prime factors so the
GCF is 1.
Greatest Common Factor
Example

Martin-Gay,
Developmental Mathematics
7
Find the GCF of each list of numbers.
1)
6, 8 and 46
6 =
2
· 3
8 =
2
· 2 · 2
46 =
2
· 23
So the GCF is 2.
2)
144, 256 and 300
144 =
2
·
2
· 2 · 3 · 3
256 =
2
·
2
· 2 · 2 · 2 · 2 · 2 · 2
300 =
2
·
2
· 3 · 5 · 5
So the GCF is
2
·
2
= 4.
Greatest Common Factor
Example

Martin-Gay,
Developmental Mathematics
8
1)
x
3
and
x
7
x
3
=
x
·
x
·
x
x
7
=
x
·
x
·
x
·
x
·
x
·
x
·
x
So the GCF is
x
·
x
·
x
=
x
3
•
6
x
5
and 4
x
3
6
x
5
=
2
· 3 ·
x
·
x
·
x
4
x
3
=
2
· 2 ·
x
·
x
·
x
So the GCF is
2
·
x
·
x
·
x
= 2
x
3
Find the GCF of each list of terms.
Greatest Common Factor
Example

Martin-Gay,
Developmental Mathematics
9
Find the GCF of the following list of terms.
a
3
b
2
,
a
2
b
5
and
a
4
b
7
a
3
b
2
=
a
·
a
·
a
·
b
·
b
a
2
b
5
=
a
·
a
·
b
·
b
·
b
·
b
·
b
a
4
b
7
=
a
·
a
·
a
·
a
·
b
·
b
·
b
·
b
·
b
·
b
·
b
So the GCF is
a
·
a
·
b
·
b
=
a
2
b
2
Notice that the GCF of terms containing variables will use the
smallest exponent found amongst the individual terms for each
variable.
Greatest Common Factor
Example

Martin-Gay,
Developmental Mathematics
10
The first step in factoring a polynomial is to
find the GCF of all its terms.
Then we write the polynomial as a product by
factoring out
the GCF from all the terms.

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