Section 7.4 Additional Factoring Techniques
Objectives
In this section, you will learn to:
•
•
•
To successfully complete this section,
you need to understand:
Factor trinomials when a = 1.
Recognize perfect square trinomials.
Factor the difference of squares.
•
•
•
•
•
Multiplying binomials (6.6)
Conjugates (6.6)
The Factor Game (7.2)
Factor by grouping (7.2)
Factoring trinomials (7.3)
INTRODUCTION
In this section, we continue our discussion on factoring polynomials by looking at special types of
trinomials and binomials. Let’s first review what we know about factoring:
(1) To factor means to write a polynomial in a factored form, one in which the main operation is
multiplication. A factored form is the product of two polynomials.
We can verify if a polynomial is in the correct factored form by multiplying the factors (often
done as work off to the side or on scratch paper).
For example, when we factor the trinomial 2x2 + 7x + 6 as (2x + 3)(x + 2), we can verify
that this is the correct factored form by multiplying the factors:
(2x + 3)(x + 2)
= 2x2 + 4x + 3x + 6
Distribute.
Combine like terms.
= 2x2 + 7x + 6 ✔ Correct!
(2) Factoring is the reverse of the distributive property. In a polynomial, we can look for a
common monomial factor in the terms, a monomial that may have been distributed to each
term.
For example, in the polynomial 2ab3 + 2ac4, we can see a common factor of 2a. This means
that 2a may have been distributed to the terms. We can factor out, or extract, this common
monomial to get a factored form:
2ab3 + 2ac4 = 2a(b3 + c4)
Factoring Trinomials, a = 1
page 7.4 - 1
(3) For a trinomial, if the corresponding Factor Game has a winning combination, then the
trinomial is guaranteed to factor. If it has no winning combination, then trinomial is not
factorable.
For example, the Factor Game for 2x2 + 7x + 6 is
Product = 2

 Sum = +7
· 6 = 12
and it has a winning combination: +3 and +4. So, 2x2 + 7x + 6 is factorable.
Whereas, the Factor Game for 4x2 + 2x + 5 is
Product = 4

 Sum = +2
· 5 = 20
has no winning combination, so 4x2 + 2x + 5 is not factorable.
Important Note:
The commutative property allows the factors to be written in either order.
For instance, in Example 2(b), x2 + 2x – 24 can be factored as either
(x + 6)(x – 4)
or as
(x – 4)(x + 6).
TRINOMIALS IN THE FORM x2 + bx + c
In Section 7.3, we discussed trinomials in the form ax2 + bx + c, where a > 1. In this section, we
explore the factoring similarities and differences of a trinomial in which a = 1: x2 + bx + c. In this
simplified form, we still use the Factor Game, and we can still use factor by grouping, as demonstrated
in Example 1.
After Example 1, however, we will discuss a “one-step” technique that works well only for factoring
trinomials of the form x2 + bx + c. (Note: this one-step technique will never work for factoring
trinomials with a lad coefficient greater than 1.)
Example 1:
Factor each trinomial.
a)
Procedure:
x2 + 5x – 36
b)
x2 – 16x + 60
Use the winning combination of the Factor Game to rewrite the trinomial as a
quadrinomial. Then use factor by grouping to write it in a factored form.
Factoring Trinomials, a = 1
page 7.4 - 2
Answer:
a)
x2 + 5x – 36
Product = 1 · ( -36) = -36 and Sum = + 5.
The winning combination is -4 and +9.
Use the winning combination to split the middle term into – 4x + 9x.
= x2 – 4x + 9x – 36
Show the groupings.
= (x2 – 4x ) + ( 9x – 36)
Extract the common monomial
factor from each group.
= x(x – 4) + 9(x – 4)
Factor out the common binomial factor, (x – 4).
= (x + 9)(x – 4)
b)
x2 – 16x + 60
Product = 1 · (60) = 60 and Sum = -16
The winning combination is -10 and -6.
Use the winning combination to split the middle term into – 10x – 6x.
= x2 – 10x – 6x + 60
Show the groupings.
= (x2 – 10x) + ( -6x + 60)
Extract the common monomial
factor from each group.
= x(x – 10) + -6(x – 10)
Factor out the common binomial factor, (x – 10).
= (x – 6)(x – 10)
You Try It 1
a)
Factor each trinomial. Use Example 1 as a guide.
x2 + 12x + 20
Factoring Trinomials, a = 1
b)
x2 – 13x + 30
page 7.4 - 3
c)
x2 – 3x – 40
d)
x2 – 8x + 12
Let’s make a connection between the winning combination of the Factor Game and the binomial factors
in Example 1:
In part a), the winning combination is +9 and -4 ,
and the binomial factors are (x + 9) and (x – 4):
x2 + 5x – 36 = (x + 9)(x – 4)
In part b), the winning combination is -6 and -10 ,
and the binomial factors are (x – 6) and (x – 10):
x2 – 16x + 60 = (x – 6)(x – 10)
Notice that, in each case, the constants of binomial factors are the exact values of the winning
combination. Notice also that the first term in each binomial is just x.
Next look at your work in You Try It 1. First check your answers for accuracy, then look at the
winning combination of the Factor Game and at the constant in each binomial. If done correctly,
they should match in number and sign.
The point is this: if a trinomial of the form x2 + bx + c is factorable—there is a winning combination
to its Factor Game—then
1)
The first term of each binomial factor is x, as in (x
2)
the constant terms of the binomials are the factor pair in the winning combination.
Factoring Trinomials, a = 1
)(x
),
and
page 7.4 - 4
If a trinomial of the form x2 + bx + c is factorable, why must the first term of
each binomial be x, as in (x )(x
)?
Think About It 1
Caution:
This “one-step” technique of using the winning combination for the binomial
constants works only for factoring trinomials with a lead coefficient of 1, as in
x2 + bx + c.
This one-step technique will never work for a trinomial with a lead coefficient
greater than 1, as in ax2 + bx + c.
Example 2 outlines the one-step technique for factoring trinomials of the form x2 + bx + c.
Example 2:
Factor each trinomial using the Factor Game.
a)
x2 – 7x + 10
b)
x2 + 2x – 24
Procedure:
First, identify the product and sum numbers, and play the Factor Game.
Second, prepare the binomial factors by setting up the parentheses and the variables.
Third, place the winning combination of the Factor Game as the constant terms in the
binomials.
Fourth, verify that the result is the correct factored form.
Answer:
a)
x2 – 7x + 10
Product = 1 · 10 = 10 and Sum = -7.
The winning combination is -2 and -5.
=
=
(x
)(x
(x
– 5)(x – 2)
)
The two steps shown at left are really one step
 in actual work. The first step shows the start,
and the second step the completion.
Verify:
Factoring Trinomials, a = 1
(x
– 5)(x – 2)
=
=
x2 – 2x – 5x + 10
x2 – 7x + 10 ✔
page 7.4 - 5
b)
x2 + 2x – 24
Product = 1 · (-24) = -24 and Sum = +2
The winning combination is +6 and -4.
=
(x
)(x
)
=
(x
+ 6)(x – 4)
Verify:
(x
+ 6)(x – 4)
=
=
x2 – 4x + 6x – 24
x2 + 2x – 24 ✔
Also, it is still possible to have a trinomial that is prime. This will happen if there are no solutions to the
Factor Game.
Example 3:
Factor x2 + 5x + 30, if possible.
x2
+ 5x + 30
is prime
Product = 1 · 30 = 30 
 no solution!

Sum = + 5
Because the factor game has no solution, the trinomial is prime
and
cannot be factored.
Because we get such quick factoring results when the leading coefficient is 1, we might call these simple
trinomials. As for those trinomials with a leading coefficient greater than 1, we might call them multistep trinomials.
You Try It 2
Factor each trinomial. If a trinomial is not factorable, write prime. Also, determine
whether it is a simple trinomial or a multi-step trinomial. Use Examples 2 and 3 as
guides.
a)
x2 + 15x + 36
b)
x2 – 11x + 24
c)
x2 + 3x – 40
d)
5x2 – 7x – 6
Factoring Trinomials, a = 1
page 7.4 - 6
e)
x2 + 4x – 18
f)
x2 – 8x + 16
PERFECT SQUARE TRINOMIALS
Consider the factoring of You Try It 2, part f): x2 – 8x + 16 = (x – 4)(x – 4)
Notice that the binomial factors are exactly the same. Because they are the same, we can write the
factored form as (x – 4)2. In this way, we could say that
•
because 49 = 72, it must be that 49 is a perfect square;
•
because x2 – 8x + 16 = (x – 4)2, x2 – 8x + 16 is also a perfect square.
x2 – 8x + 16 is called a perfect square trinomial.
If a trinomial is a perfect square, such as x2 – 8x + 16, it is appropriate to write the factored form as
the square of a binomial, (x – 4)2.
Here are some examples of perfect square trinomials.
x2 + 14x + 49
a)
x2 – 10x + 25
b)
=
(x
+ 7)(x + 7)
=
(x
– 5)(x – 5)
=
(x
+ 7)2
=
(x
– 5)2
x2 – 12x + 36
c)
x2 + 6x + 9
d)
=
(x
– 6)(x – 6)
=
(x
+ 3)(x + 3)
=
(x
– 6) 2
=
(x
+ 3)2
It is relatively easy to identify perfect square trinomials using the Factor Game. This is because,
whenever the factor pair of the winning combination is the same number, the trinomial is a perfect
square trinomial, as demonstrated in Example 4.
Factoring Trinomials, a = 1
page 7.4 - 7
Example 4:
a)
We already know that these trinomials are perfect square trinomials. Show that the
winning combination of the Factor Game is a factor pair of the same number.
x2 + 14x + 49
Procedure:
Answer:
Caution:
You Try It 3
b)
x2 – 10x + 25
c)
x2 – 12x + 36
d)
x2 + 6x + 9
Play the Factor Game for each trinomial. The winning combination should be a factor
pair of the same number.
a)
Product = + 49 
 + 7 and + 7

Sum = + 14
c)
Product = + 36 


Sum = -12
-6 and -6
b)
Product = + 25 


Sum = -10
d)
Product = + 9 
 + 3 and + 3

Sum = + 6
-5 and -5
If the numbers in the factor pair of the winning combination are not the same,
the trinomial is not a perfect square trinomial.
Factor each trinomial using the ideas developed in this section. If the trinomial is a
perfect square trinomial, then write the factorization as (binomial)2. If a trinomial is
not factorable, then write prime. Use the discussion above as a guide.
a)
x2 + 18x + 81
b)
x2 + 9x – 36
c)
x2 – 2x + 1
d)
9x2 – 12x + 4
e)
x2 – 10x + 16
f)
x2 + 20x + 100
Factoring Trinomials, a = 1
page 7.4 - 8
OTHER PERFECT SQUARE TRINOMIALS
As you might imagine, we can get perfect square trinomials when the lead term is greater than 1. For
example,
4x2 + 12x + 9 is a perfect square trinomial.
i)
The lead term is a perfect square, 4x2 = (2x)·(2x);
ii)
the constant term is a positive perfect square, + 9;
iii)
the Factor Game’s solutions are exactly the same; they’re both + 6:
4x2 + 12x + 9
Product = 4 · (9) = 36 and Sum = + 12.
The winning combination is +6 and +6.
The numbers of the combination are the same.
Here is the factorization of 4x2 + 12x + 9:
The original trinomial:
Split the middle term into
+ 6x + 6x:
4x2 + 12x + 9
= 4x2 + 6x + 6x + 9
Now show the groupings:
= (4x2 + 6x) + (6x + 9)
Now factor each group:
= 2x(2x + 3) + 3(2x + 3)
Now factor out (2x + 3):
= (2x + 3)(2x + 3)
= (2x + 3)2
This is a perfect square.
Here is the good news about this (or any) perfect square trinomial: if you know it’s a perfect square by
the Factor Game, then you can get the factorization in just one step.
Look at the factorization of 4x2 + 12x + 9. Notice that the square root of 4, 4 , is 2 and that
9 is 3, and that those square roots show up in the factorization: it is (2x + 3)2.
Factoring Trinomials, a = 1
page 7.4 - 9
In other words, if we know that the trinomial is a perfect square (by the Factor Game), then we need
only (carefully) place the numbers, the variable and the sign (plus or minus) into the outline
(
)2 .
Let’s take another look at factoring 4x2 + 12x + 9:
4x2 + 12x + 9
Product = 4 · (9) = 36 and Sum = + 12.
The winning combination is +6 and +6.
The numbers of the combination are the same.
The Factor Game guarantees that this will be a perfect square; the square root of 4 is 2, and the square
root of 9 is 3; the middle term is positive, so the factorization is (2x + 3)2.
Example 5:
a)
Use the Factor Game to determine that the trinomial is a perfect square, then write its
factorization as (binomial)2.
25x2 – 20x + 4
Product = + 100
Sum = -20
Solutions: -10 and -10
b)
So, it is a perfect square;
Also,
So,
25 = 5,
4
= 2
25x2 – 20x + 4 = (5x – 2)2
You Try It 4
16x2 + 8x + 1
Product = + 16
Sum = + 8
Solutions: + 4 and + 4
So, it is a perfect square;
Also,
So,
16
= 4,
1
= 1
16x2 + 8x + 1 = (4x + 1)2
First, determine if the trinomial is a perfect square. If it is, then write the
factorization as (binomial)2. If it not a perfect square, factor it using the method
shown in Section 7.3. If a trinomial is not factorable, then write prime. Use Example
5 as a guide.
a)
4x2 – 12x + 9
Factoring Trinomials, a = 1
b)
4x2 + 4x + 1
page 7.4 - 10
c)
25x2 + 30x + 9
d)
81x2 – 18x + 1
So, if the first and last terms are both perfect squares, what else is it that determines whether the whole
trinomial will be a perfect square? The easiest answer is this: If the results of the Factor Game are
exactly the same, then the trinomial is guaranteed to be a perfect square. This idea is summed up here:
A trinomial is a perfect square if
(i)
both the first term and the last term are perfect squares; and
(ii)
the results of the Factor Game are exactly the same,
THE DIFFERENCE OF SQUARES AND THE SUM OF SQUARES
Let us consider two different binomials: x2 – 25 and
common, so we can’t factor out a GCF (except 1).
x2 + 25. In each, the terms have nothing in
Notice that each term, x2 and 25, is a perfect square:
We call x2 – 25 the difference of squares, and x2 + 25 is the sum of squares.
Furthermore, we can write each binomial as a trinomial by adding in 0x as a middle term:
The difference of squares, x2 – 25, can be written as x2 + 0x – 25:
x2 + 0x – 25
Product = 1 · ( -25) = -25 and Sum = 0.
The winning combination is -5 and +5.
= (x – 5)(x + 5)
The binomials (x – 5) and (x + 5) are a pair of conjugates (first discussed in Section 6.6). So, this
example shows us that the difference of squares can be factored into a pair of conjugates.
Factoring Trinomials, a = 1
page 7.4 - 11
Likewise, the sum of squares, x2 + 25, can be written as x2 + 0x + 25:
x2 + 0x + 25
Product = 1 · (25) = 25 and Sum = 0.
There is no winning combination.
Prime!
This example shows us that the sum of squares is prime, it cannot be factored. In general,
The difference of squares, a2 – b2, factors into a pair of conjugates:
a2 – b2 = (a – b)(a + b)
The sum of squares, c2 + d2, is prime.
Note:
1. The binomial factors in the difference of squares can be written in either order.
2. The purpose of adding a middle term of 0x is just for explanation. We do not need to
add 0x to determine the factors of either of type of these binomials.
Example 6:
Factor each binomial, if possible.
a)
y2 – 4
b)
x2 + 16
w2 – 20
c)
Procedure:
Identify each as either the difference of squares or the sum of squares. The sum
of squares is prime, but the difference of squares factors into a product of
conjugates.
Answer:
a)
You Try It 5
(y + 2)(y – 2)
b)
Prime.
c) Prime. (20 is not a
perfect square.)
Factor each binomial, if possible. Use Example 6 as a guide.
a)
x2 – 49
b)
x2 + 9
c)
x2 + 81
d)
x2 – 81
d)
x2 + 100
f)
x2 – 12
Factoring Trinomials, a = 1
page 7.4 - 12
The difference of squares can come in many forms, as long as each term is a perfect square. Some
examples of perfect squares are:
49, which is (7)2
9y2, which is (3y)2 25w2, which is (5w)2
2
x4, which is (x2)
y6, which is (y3)
2
81w4, which is (9w2)
and
2
Any of these could be a term in the difference of squares, and it will still be factorable. However, the
sum of squares is prime. (There are some unusual situations when the sum of squares is not prime, but
they are beyond the scope of this textbook.)
Example 7:
a)
Factor each binomial, if possible.
9x2 – 25
b)
Procedure:
c)
9m2 + 4p2
d)
25x4 – 36y2
Identify the terms as perfect squares, and factor accordingly.
Answer:
You Try It 6
w4 – 49
a)
(3x + 5)(3x – 5)
9x2 = (3x)2 and 25 = (5)2
b)
(w2 – 7)(w2 + 7)
w4 = (w2)
c)
Prime.
9m2 + 4p2 is the sum of squares.
d)
(5x2 + 6y)(5x2 – 6y)
25x4 = (5x2)
2
and 49 = (7)2
2
and 36y2 = (6y)2
Factor each binomial, if possible. Use Example 7 as a guide.
a)
m2 – 25p2
b)
36w4 – 1
c)
16y2 + 49
d)
64a2 – 81b2
e)
x4 + 100
f)
4x6 – y2
Factoring Trinomials, a = 1
page 7.4 - 13
Section 7.4, Answers to each You Try It and Think About It
You Try It 1:
a)
c)
(x + 10)(x + 2)
(x – 8)(x + 5)
b)
d)
(x – 10)(x – 3)
(x – 6)(x – 2)
You Try It 2:
a)
c)
e)
(x + 12)(x + 3)
(x + 8)(x – 5)
prime
b)
d)
f)
(x – 8)(x – 3)
(5x + 3)(x – 2)
(x – 4)(x – 4)
You Try It 3:
a)
c)
e)
(x + 9)2
(x – 1)2
(x – 8)(x – 2)
b)
d)
f)
(x + 12)(x – 3)
(3x – 2)2
(x + 10)2
You Try It 4:
a)
c)
(2x – 3)2
(5x + 3)2
b)
d)
(2x + 1)2
(9x – 1)2
You Try It 5:
a)
c)
e)
(x + 7)(x – 7)
prime
prime
b)
d)
f)
prime
(x – 9)(x + 9)
prime
You Try It 6:
a)
c)
e)
(m + 5p)(m – 5p)
prime
prime
b)
d)
f)
(6w2 – 1)(6w2 + 1)
((8a + 9b)(8a – 9b)
(2x3 – y)(2x3 + y)
Think About It: 1. The only way to factor x2 is x · x, so the first terms of the binomials must each be x.
Factoring Trinomials, a = 1
page 7.4 - 14
Section 7.4
Focus Exercieses
Factor each trinomial, if possible. If the trinomial is a perfect square trinomial, then write its factored
form as the square of a binomial. If a trinomial is not factorable, then write prime.
1.
x2 – 12x – 32
2.
x2 – 14x + 40
3.
x2 + 15x + 36
4.
x2 – 12x + 32
5.
x2 + 8x + 12 6.
x2 – 9x + 20
7.
x2 – 11x + 36
8.
x2 – 9x – 24 9.
x2 + 2x – 48
10.
x2 – 3x – 10
11.
x2 + 3x – 18 12.
x2 – 4x – 77
13.
x2 – 16x + 64
14.
x2 + 7x + 49 15.
x2 + 5x – 50
16.
x2 – x – 42
17.
x2 – x – 30
18.
19.
x2 + 7x + 30
20.
x2 – 9x + 20 21.
x2 + 10x – 24
22.
x2 – 9x – 36
23.
x2 + 13x + 36
24.
x2 – 4x – 20
25.
x2 + 8x + 16
26.
x2 – 10x + 25
27.
x2 – 12x + 36
28.
x2 + 6x + 9
29.
x2 + 13x + 36
30.
x2 + 18x + 81
31.
x2 + 24x – 25
32.
x2 + 12x – 36
33.
x2 + 10x + 18
34.
x2 + 22x + 121
35.
x2 – 2x + 1 36.
x2 – 4x + 4
37.
x2 – 15x + 36
38.
x2 – 8x + 64 39.
6x2 + 10x – 4
40.
8x2 + 6x – 5
41.
9x2 + 9x + 2
42.
43.
16x2 + 8x + 1
44.
9x2 + 6x + 1 45.
4x2 – 20x + 25
46.
9x2 – 15x + 4
47.
64x2 – 16x + 1
48.
25x2 – 5x + 1
x2 – 13x – 30
5x2 – 4x – 12
Factor each binomial, if possible.
49.
x2 – 36
50.
p2 – 81
51.
y2 + 16
52.
a2 + 4
53.
16w2 – 1
54.
9x2 – 4
Factoring Trinomials, a = 1
page 7.4 - 15
55.
81a2 – 25b2
56.
100x2 – y2
57.
25y2 + 16
58.
49a2 + 16b2
59.
x4 – 25
60.
w4 – 49
61.
4c4 – d2
62.
100x4 – 81y2
63.
x6 – 49
64.
a6 – 16
65.
25x6 – 64w4
66.
81m6 – 16p2
Think Outside the Box.
Each of these can be factored more than once. In other words, more than one technique (or even the
same technique) can be applied in different steps of the factoring process.
67.
x3 – 7x2 – 8x
68.
5x2 – 15x – 20
69.
12x3 – 27x
70.
y4 – 81
71.
- 6x3 – 12x2 + 18x
72.
- 6x2 + 4x + 10
Factoring Trinomials, a = 1
page 7.4 - 16