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Elementary Algebra

7.2 Factor Quadratic Trinomials with Leading Coefficient 1

Elementary Algebra7.2 Factor Quadratic Trinomials with Leading Coefficient 1
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope–Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solve Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Quadratic Trinomials with Leading Coefficient 1
    4. 7.3 Factor Quadratic Trinomials with Leading Coefficient Other than 1
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

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.

This figure shows the steps of multiplying the factors (x + 2) times (x + 3). The multiplying is completed using FOIL to demonstrate. The first term is x squared and is below F. The second term is 3 x below “O”. The third term is 2 x below “I”. The fourth term is 6 below L. The simplified product is then given as x 2 plus 5 x + 6.

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: t211t+28t211t+28.

Try It 7.39

Factor: u29u+18u29u+18.

Try It 7.40

Factor: y216y+63y216y+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+4z5z2+4z5.

Try It 7.41

Factor: h2+4h12h2+4h12.

Try It 7.42

Factor: k2+k20k2+k20.

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

Example 7.22

Factor: z24z5z24z5.

Try It 7.43

Factor: x24x12x24x12.

Try It 7.44

Factor: y2y20y2y20.

Example 7.23

Factor: q22q15q22q15.

Try It 7.45

Factor: r23r40r23r40.

Try It 7.46

Factor: s23s10s23s10.

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: y26y+15y26y+15.

Try It 7.47

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

Try It 7.48

Factor: n210n+12n210n+12.

Example 7.25

Factor: 2x+x2482x+x248.

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+6x26x+8(x+2)(x+3)(x4)(x2)same signssame signsbpositivebnegativem,npositivem,nnegativex2+5x+6x26x+8(x+2)(x+3)(x4)(x2)same signssame signs

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

x2+x12x22x15(x+4)(x3)(x5)(x+3)opposite signsopposite signsx2+x12x22x15(x+4)(x3)(x5)(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: r28rs9s2r28rs9s2.

Try It 7.53

Factor: a211ab+10b2a211ab+10b2.

Try It 7.54

Factor: m213mn+12n2m213mn+12n2.



Example 7.28

Factor: u29uv12v2u29uv12v2.

Try It 7.55

Factor: x27xy10y2x27xy10y2.

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.

x2+4x+3x2+4x+3

64.

y2+8y+7y2+8y+7

65.

m2+12m+11m2+12m+11

66.

b2+14b+13b2+14b+13

67.

a2+9a+20a2+9a+20

68.

m2+7m+12m2+7m+12

69.

p2+11p+30p2+11p+30

70.

w2+10x+21w2+10x+21

71.

n2+19n+48n2+19n+48

72.

b2+14b+48b2+14b+48

73.

a2+25a+100a2+25a+100

74.

u2+101u+100u2+101u+100

75.

x28x+12x28x+12

76.

q213q+36q213q+36

77.

y218y+45y218y+45

78.

m213m+30m213m+30

79.

x28x+7x28x+7

80.

y25y+6y25y+6

81.

p2+5p6p2+5p6

82.

n2+6n7n2+6n7

83.

y26y7y26y7

84.

v22v3v22v3

85.

x2x12x2x12

86.

r22r8r22r8

87.

a23a28a23a28

88.

b213b30b213b30

89.

w25w36w25w36

90.

t23t54t23t54

91.

x2+x+5x2+x+5

92.

x23x9x23x9

93.

86x+x286x+x2

94.

7x+x2+67x+x2+6

95.

x21211xx21211x

96.

−1110x+x2−1110x+x2

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.

p2+3pq+2q2p2+3pq+2q2

98.

m2+6mn+5n2m2+6mn+5n2

99.

r2+15rs+36s2r2+15rs+36s2

100.

u2+10uv+24v2u2+10uv+24v2

101.

m212mn+20n2m212mn+20n2

102.

p216pq+63q2p216pq+63q2

103.

x22xy80y2x22xy80y2

104.

p28pq65q2p28pq65q2

105.

m264mn65n2m264mn65n2

106.

p22pq35q2p22pq35q2

107.

a2+5ab24b2a2+5ab24b2

108.

r2+3rs28s2r2+3rs28s2

109.

x23xy14y2x23xy14y2

110.

u28uv24v2u28uv24v2

111.

m25mn+30n2m25mn+30n2

112.

c27cd+18d2c27cd+18d2

Mixed Practice

In the following exercises, factor each expression.

113.

u212u+36u212u+36

114.

w2+4w32w2+4w32

115.

x214x32x214x32

116.

y2+41y+40y2+41y+40

117.

r220rs+64s2r220rs+64s2

118.

x216xy+64y2x216xy+64y2

119.

k2+34k+120k2+34k+120

120.

m2+29m+120m2+29m+120

121.

y2+10y+15y2+10y+15

122.

z23z+28z23z+28

123.

m2+mn56n2m2+mn56n2

124.

q229qr96r2q229qr96r2

125.

u217uv+30v2u217uv+30v2

126.

m231mn+30n2m231mn+30n2

127.

c28cd+26d2c28cd+26d2

128.

r2+11rs+36s2r2+11rs+36s2

Everyday Math

129.

Consecutive integers Deirdre is thinking of two consecutive integers whose product is 56. The trinomial x2+x56x2+x56 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+x182x2+x182 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 x2x20x2x20 as (x+5)(x4)(x+5)(x4). Bill factored it as (x+4)(x5)(x+4)(x5). Phil factored it as (x5)(x4)(x5)(x4). 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.

This table has the following statements all to be preceded by “I can…”. The first is “factor trinomials of the form x ^ 2 +b x + c”. The second is “factor trinomials of the form x^2 + b x y + c y ^ 2”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.

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

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