Examples:
Factor the followings by grouping.
1.
3x(x + y) + 2(x + y)
2.
3x
2
(x
–
y) + 6x(x
–
y) + 9(x
–
y)
3.
ax
–
ay + bx
–
by
4.
2x
3
+ 3x
2
–
6x + 9
5.
x
2
–
6yz + 2xy
–
3xz
8

5.2: Factoring Special Polynomials
Objectives:
5.2.1: Factor the difference of squares
5.2.2: Factor the sum and difference of cubes
9

5.2.1: Factor the Difference of Squares
The product of a sum and difference of two terms such as
(a + b)(a
–
b) = a
2
–
b
2
This also means that a binomial of the form a
2
–
b
2
is called
the
difference of squares,
and its factors a + b and a
–
b
Examples:
Factor the followings
1.
x
2
–
16
2.
4a
2
–
9
3.
25a
2
–
16b
2
4.
32x
2
y
–
18y
3
5.
m
4
–
81n
4
10

5.2.1: Factor the Square of a Binomial
Squaring
a binomial always results in
three
terms.
Examples:
(x + y)
2
= (x + y)(x + y) = x
2
+ xy + xy + y
2
= x
2
+ 2xy + y
2
(x
–
y)
2
= (x
–
y)(x
–
y) = x
2
–
xy
–
xy + y
2
= x
2
–
2xy + y
2
Examples:
Factor the followings
1.
x
2
–
6xy + 9y
2
2.
x
3
y
+ 6x
2
y
2
+ 9xy
3
11

5.2.2: Factor the Sum and Difference of Cubes
Two additional factoring patterns are the sum and difference
of cubes.
Examples:
Factor the followings
1.
x
3
+ 27
2.
8w
3
–
27z
3
3.
5a
3
b
–
40b
4
Examples:
y =9x
2
+ 15x
1.
Find y if x = 1.
2.
Rewrite the polynomial equation by factoring the right side.
12

5.3: Factoring Trinomials: Trial and Error
Objectives:
5.3.1: Factor a trinomial of the form x
2
+ bx + c
5.3.2: Factor a trinomial of the form ax
2
+ bx + c
5.3.3: Completely factor a trinomial
13

5.3.1: Factor a Trinomial of the Form x
2
+ bx + c
Recall that the product of two binomials may be a trinomial
of the form ax
2
+ bx + c
This suggests that some trinomials may be factored as the
product of two binomials.
One process for factoring a trinomial into a product of two
binomials is called
trial and error
.
14

5.3.1: Factor a Trinomial of the Form x
2
+ bx + c
15

5.3.1: Factor a Trinomial of the Form x
2
+ bx + c
Examples:
Factor the following trinomials.
1.
x
2
+ 7x + 10
2.
x
2
–
9x + 14
3.
x
2
+ 4x
–
12
16

5.3.2: Factor a Trinomial of the Form ax
2
+ bx + c
To factor a trinomial of the form ax
2
+ bx + c, we must
consider binomial factors of (_x + _) (_x + _)
17

5.3.2: Factor a Trinomial of the Form ax
2
+ bx + c
Examples:
Factor the following trinomials.

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- Fall '18
- jane
- Accounting, Quadratic equation, Elementary algebra