Elementary Algebra

### Learning Objectives

By the end of this section, you will be able to:

• Factor trinomials of the form $x2+bx+cx2+bx+c$
• Factor trinomials of the form $x2+bxy+cy2x2+bxy+cy2$

### Be Prepared 7.2

Before you get started, take this readiness quiz.

1. Multiply: $(x+4)(x+5).(x+4)(x+5).$
If you missed this problem, review Example 6.38.
2. Simplify: $−9+(−6)−9+(−6)$ $−9+6.−9+6.$
If you missed this problem, review Example 1.37.
3. Simplify: $−9(6)−9(6)$ $−9(−6).−9(−6).$
If you missed this problem, review Example 1.46.
4. Simplify: $|−5||−5|$ $|3|.|3|.$
If you missed this problem, review Example 1.33.

### Factor Trinomials of the Form x2 + bx + c

You have already learned how to multiply binomials using FOIL. Now you’ll need to “undo” this multiplication—to start with the product and end up with the factors. Let’s look at an example of multiplying binomials to refresh your memory.

To factor the trinomial means to start with the product, $x2+5x+6x2+5x+6$, and end with the factors, $(x+2)(x+3)(x+2)(x+3)$. You need to think about where each of the terms in the trinomial came from.

The first term came from multiplying the first term in each binomial. So to get $x2x2$ in the product, each binomial must start with an x.

$x2+5x+6(x)(x)x2+5x+6(x)(x)$

The last term in the trinomial came from multiplying the last term in each binomial. So the last terms must multiply to 6.

What two numbers multiply to 6?

The factors of 6 could be 1 and 6, or 2 and 3. How do you know which pair to use?

Consider the middle term. It came from adding the outer and inner terms.

So the numbers that must have a product of 6 will need a sum of 5. We’ll test both possibilities and summarize the results in Table 7.1—the table will be very helpful when you work with numbers that can be factored in many different ways.

Factors of $66$ Sum of factors
$1,61,6$ $1+6=71+6=7$
$2,32,3$ $2+3=52+3=5$
Table 7.1

We see that 2 and 3 are the numbers that multiply to 6 and add to 5. So we have the factors of $x2+5x+6x2+5x+6$. They are $(x+2)(x+3)(x+2)(x+3)$.

$x2+5x+6product(x+2)(x+3)factorsx2+5x+6product(x+2)(x+3)factors$

You should check this by multiplying.

Looking back, we started with $x2+5x+6x2+5x+6$, which is of the form $x2+bx+cx2+bx+c$, where $b=5b=5$ and $c=6c=6$. We factored it into two binomials of the form $(x+m)and(x+n)(x+m)and(x+n)$.

$x2+5x+6x2+bx+c(x+2)(x+3)(x+m)(x+n)x2+5x+6x2+bx+c(x+2)(x+3)(x+m)(x+n)$

To get the correct factors, we found two numbers m and n whose product is c and sum is b.

### Example 7.17

#### How to Factor Trinomials of the Form $x2+bx+cx2+bx+c$

Factor: $x2+7x+12x2+7x+12$.

### Try It 7.33

Factor: $x2+6x+8x2+6x+8$.

### Try It 7.34

Factor: $y2+8y+15y2+8y+15$.

Let’s summarize the steps we used to find the factors.

### How To

#### Factor trinomials of the form $x2+bx+cx2+bx+c$.

1. Step 1. Write the factors as two binomials with first terms x: $(x)(x)(x)(x)$.
2. Step 2. Find two numbers m and n that
Multiply to c, $m·n=cm·n=c$
Add to b, $m+n=bm+n=b$
3. Step 3. Use m and n as the last terms of the factors: $(x+m)(x+n)(x+m)(x+n)$.
4. Step 4. Check by multiplying the factors.

### Example 7.18

Factor: $u2+11u+24u2+11u+24$.

### Try It 7.35

Factor: $q2+10q+24q2+10q+24$.

### Try It 7.36

Factor: $t2+14t+24t2+14t+24$.

### Example 7.19

Factor: $y2+17y+60y2+17y+60$.

### Try It 7.37

Factor: $x2+19x+60x2+19x+60$.

### Try It 7.38

Factor: $v2+23v+60v2+23v+60$.

#### Factor Trinomials of the Form x2 + bx + c with b Negative, c Positive

In the examples so far, all terms in the trinomial were positive. What happens when there are negative terms? Well, it depends which term is negative. Let’s look first at trinomials with only the middle term negative.

Remember: To get a negative sum and a positive product, the numbers must both be negative.

Again, think about FOIL and where each term in the trinomial came from. Just as before,

• the first term, $x2x2$, comes from the product of the two first terms in each binomial factor, x and y;
• the positive last term is the product of the two last terms
• the negative middle term is the sum of the outer and inner terms.

How do you get a positive product and a negative sum? With two negative numbers.

### Example 7.20

Factor: $t2−11t+28t2−11t+28$.

### Try It 7.39

Factor: $u2−9u+18u2−9u+18$.

### Try It 7.40

Factor: $y2−16y+63y2−16y+63$.

#### Factor Trinomials of the Form $x2+bx+cx2+bx+c$ with c Negative

Now, what if the last term in the trinomial is negative? Think about FOIL. The last term is the product of the last terms in the two binomials. A negative product results from multiplying two numbers with opposite signs. You have to be very careful to choose factors to make sure you get the correct sign for the middle term, too.

Remember: To get a negative product, the numbers must have different signs.

### Example 7.21

Factor: $z2+4z−5z2+4z−5$.

### Try It 7.41

Factor: $h2+4h−12h2+4h−12$.

### Try It 7.42

Factor: $k2+k−20k2+k−20$.

Let’s make a minor change to the last trinomial and see what effect it has on the factors.

### Example 7.22

Factor: $z2−4z−5z2−4z−5$.

### Try It 7.43

Factor: $x2−4x−12x2−4x−12$.

### Try It 7.44

Factor: $y2−y−20y2−y−20$.

### Example 7.23

Factor: $q2−2q−15q2−2q−15$.

### Try It 7.45

Factor: $r2−3r−40r2−3r−40$.

### Try It 7.46

Factor: $s2−3s−10s2−3s−10$.

Some trinomials are prime. The only way to be certain a trinomial is prime is to list all the possibilities and show that none of them work.

### Example 7.24

Factor: $y2−6y+15y2−6y+15$.

### Try It 7.47

Factor: $m2+4m+18m2+4m+18$.

### Try It 7.48

Factor: $n2−10n+12n2−10n+12$.

### Example 7.25

Factor: $2x+x2−482x+x2−48$.

### Try It 7.49

Factor: $9m+m2+189m+m2+18$.

### Try It 7.50

Factor: $−7n+12+n2−7n+12+n2$.

Let’s summarize the method we just developed to factor trinomials of the form $x2+bx+cx2+bx+c$.

### How To

#### Factor trinomials.

When we factor a trinomial, we look at the signs of its terms first to determine the signs of the binomial factors.

$x2+bx+c(x+m)(x+n)x2+bx+c(x+m)(x+n)$

When c is positive, m and n have the same sign.

$bpositivebnegativem,npositivem,nnegativex2+5x+6x2−6x+8(x+2)(x+3)(x−4)(x−2)same signssame signsbpositivebnegativem,npositivem,nnegativex2+5x+6x2−6x+8(x+2)(x+3)(x−4)(x−2)same signssame signs$

When c is negative, m and n have opposite signs.

$x2+x−12x2−2x−15(x+4)(x−3)(x−5)(x+3)opposite signsopposite signsx2+x−12x2−2x−15(x+4)(x−3)(x−5)(x+3)opposite signsopposite signs$

Notice that, in the case when m and n have opposite signs, the sign of the one with the larger absolute value matches the sign of b.

### Factor Trinomials of the Form x2 + bxy + cy2

Sometimes you’ll need to factor trinomials of the form $x2+bxy+cy2x2+bxy+cy2$ with two variables, such as $x2+12xy+36y2.x2+12xy+36y2.$ The first term, $x2x2$, is the product of the first terms of the binomial factors, $x·xx·x$. The $y2y2$ in the last term means that the second terms of the binomial factors must each contain y. To get the coefficients b and c, you use the same process summarized in the previous objective.

### Example 7.26

Factor: $x2+12xy+36y2x2+12xy+36y2$.

### Try It 7.51

Factor: $u2+11uv+28v2u2+11uv+28v2$.

### Try It 7.52

Factor: $x2+13xy+42y2x2+13xy+42y2$.

### Example 7.27

Factor: $r2−8rs−9s2r2−8rs−9s2$.

### Try It 7.53

Factor: $a2−11ab+10b2a2−11ab+10b2$.

### Try It 7.54

Factor: $m2−13mn+12n2m2−13mn+12n2$.

### Example 7.28

Factor: $u2−9uv−12v2u2−9uv−12v2$.

### Try It 7.55

Factor: $x2−7xy−10y2x2−7xy−10y2$.

### Try It 7.56

Factor: $p2+15pq+20q2p2+15pq+20q2$.

### Section 7.2 Exercises

#### Practice Makes Perfect

Factor Trinomials of the Form $x2+bx+cx2+bx+c$

In the following exercises, factor each trinomial of the form $x2+bx+cx2+bx+c$.

63.

$x 2 + 4 x + 3 x 2 + 4 x + 3$

64.

$y 2 + 8 y + 7 y 2 + 8 y + 7$

65.

$m 2 + 12 m + 11 m 2 + 12 m + 11$

66.

$b 2 + 14 b + 13 b 2 + 14 b + 13$

67.

$a 2 + 9 a + 20 a 2 + 9 a + 20$

68.

$m 2 + 7 m + 12 m 2 + 7 m + 12$

69.

$p 2 + 11 p + 30 p 2 + 11 p + 30$

70.

$w 2 + 10 x + 21 w 2 + 10 x + 21$

71.

$n 2 + 19 n + 48 n 2 + 19 n + 48$

72.

$b 2 + 14 b + 48 b 2 + 14 b + 48$

73.

$a 2 + 25 a + 100 a 2 + 25 a + 100$

74.

$u 2 + 101 u + 100 u 2 + 101 u + 100$

75.

$x 2 − 8 x + 12 x 2 − 8 x + 12$

76.

$q 2 − 13 q + 36 q 2 − 13 q + 36$

77.

$y 2 − 18 y + 45 y 2 − 18 y + 45$

78.

$m 2 − 13 m + 30 m 2 − 13 m + 30$

79.

$x 2 − 8 x + 7 x 2 − 8 x + 7$

80.

$y 2 − 5 y + 6 y 2 − 5 y + 6$

81.

$p 2 + 5 p − 6 p 2 + 5 p − 6$

82.

$n 2 + 6 n − 7 n 2 + 6 n − 7$

83.

$y 2 − 6 y − 7 y 2 − 6 y − 7$

84.

$v 2 − 2 v − 3 v 2 − 2 v − 3$

85.

$x 2 − x − 12 x 2 − x − 12$

86.

$r 2 − 2 r − 8 r 2 − 2 r − 8$

87.

$a 2 − 3 a − 28 a 2 − 3 a − 28$

88.

$b 2 − 13 b − 30 b 2 − 13 b − 30$

89.

$w 2 − 5 w − 36 w 2 − 5 w − 36$

90.

$t 2 − 3 t − 54 t 2 − 3 t − 54$

91.

$x 2 + x + 5 x 2 + x + 5$

92.

$x 2 − 3 x − 9 x 2 − 3 x − 9$

93.

$8 − 6 x + x 2 8 − 6 x + x 2$

94.

$7 x + x 2 + 6 7 x + x 2 + 6$

95.

$x 2 − 12 − 11 x x 2 − 12 − 11 x$

96.

$−11 − 10 x + x 2 −11 − 10 x + x 2$

Factor Trinomials of the Form $x2+bxy+cy2x2+bxy+cy2$

In the following exercises, factor each trinomial of the form $x2+bxy+cy2x2+bxy+cy2$.

97.

$p 2 + 3 p q + 2 q 2 p 2 + 3 p q + 2 q 2$

98.

$m 2 + 6 m n + 5 n 2 m 2 + 6 m n + 5 n 2$

99.

$r 2 + 15 r s + 36 s 2 r 2 + 15 r s + 36 s 2$

100.

$u 2 + 10 u v + 24 v 2 u 2 + 10 u v + 24 v 2$

101.

$m 2 − 12 m n + 20 n 2 m 2 − 12 m n + 20 n 2$

102.

$p 2 − 16 p q + 63 q 2 p 2 − 16 p q + 63 q 2$

103.

$x 2 − 2 x y − 80 y 2 x 2 − 2 x y − 80 y 2$

104.

$p 2 − 8 p q − 65 q 2 p 2 − 8 p q − 65 q 2$

105.

$m 2 − 64 m n − 65 n 2 m 2 − 64 m n − 65 n 2$

106.

$p 2 − 2 p q − 35 q 2 p 2 − 2 p q − 35 q 2$

107.

$a 2 + 5 a b − 24 b 2 a 2 + 5 a b − 24 b 2$

108.

$r 2 + 3 r s − 28 s 2 r 2 + 3 r s − 28 s 2$

109.

$x 2 − 3 x y − 14 y 2 x 2 − 3 x y − 14 y 2$

110.

$u 2 − 8 u v − 24 v 2 u 2 − 8 u v − 24 v 2$

111.

$m 2 − 5 m n + 30 n 2 m 2 − 5 m n + 30 n 2$

112.

$c 2 − 7 c d + 18 d 2 c 2 − 7 c d + 18 d 2$

Mixed Practice

In the following exercises, factor each expression.

113.

$u 2 − 12 u + 36 u 2 − 12 u + 36$

114.

$w 2 + 4 w − 32 w 2 + 4 w − 32$

115.

$x 2 − 14 x − 32 x 2 − 14 x − 32$

116.

$y 2 + 41 y + 40 y 2 + 41 y + 40$

117.

$r 2 − 20 r s + 64 s 2 r 2 − 20 r s + 64 s 2$

118.

$x 2 − 16 x y + 64 y 2 x 2 − 16 x y + 64 y 2$

119.

$k 2 + 34 k + 120 k 2 + 34 k + 120$

120.

$m 2 + 29 m + 120 m 2 + 29 m + 120$

121.

$y 2 + 10 y + 15 y 2 + 10 y + 15$

122.

$z 2 − 3 z + 28 z 2 − 3 z + 28$

123.

$m 2 + m n − 56 n 2 m 2 + m n − 56 n 2$

124.

$q 2 − 29 q r − 96 r 2 q 2 − 29 q r − 96 r 2$

125.

$u 2 − 17 u v + 30 v 2 u 2 − 17 u v + 30 v 2$

126.

$m 2 − 31 m n + 30 n 2 m 2 − 31 m n + 30 n 2$

127.

$c 2 − 8 c d + 26 d 2 c 2 − 8 c d + 26 d 2$

128.

$r 2 + 11 r s + 36 s 2 r 2 + 11 r s + 36 s 2$

#### Everyday Math

129.

Consecutive integers Deirdre is thinking of two consecutive integers whose product is 56. The trinomial $x2+x−56x2+x−56$ describes how these numbers are related. Factor the trinomial.

130.

Consecutive integers Deshawn is thinking of two consecutive integers whose product is 182. The trinomial $x2+x−182x2+x−182$ describes how these numbers are related. Factor the trinomial.

#### Writing Exercises

131.

Many trinomials of the form $x2+bx+cx2+bx+c$ factor into the product of two binomials $(x+m)(x+n)(x+m)(x+n)$. Explain how you find the values of m and n.

132.

How do you determine whether to use plus or minus signs in the binomial factors of a trinomial of the form $x2+bx+cx2+bx+c$ where $bb$ and $cc$ may be positive or negative numbers?

133.

Will factored $x2−x−20x2−x−20$ as $(x+5)(x−4)(x+5)(x−4)$. Bill factored it as $(x+4)(x−5)(x+4)(x−5)$. Phil factored it as $(x−5)(x−4)(x−5)(x−4)$. Who is correct? Explain why the other two are wrong.

134.

Look at Example 7.19, where we factored $y2+17y+60y2+17y+60$. We made a table listing all pairs of factors of 60 and their sums. Do you find this kind of table helpful? Why or why not?

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After reviewing this checklist, what will you do to become confident for all goals?

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