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Intermediate Algebra 2e

6.3 Factor Special Products

Intermediate Algebra 2e6.3 Factor Special Products
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. 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
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

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

  • Factor perfect square trinomials
  • Factor differences of squares
  • Factor sums and differences of cubes
Be Prepared 6.7

Before you get started, take this readiness quiz.

Simplify: (3x2)3.(3x2)3.
If you missed this problem, review Example 5.18.

Be Prepared 6.8

Multiply: (m+4)2.(m+4)2.
If you missed this problem, review Example 5.32.

Be Prepared 6.9

Multiply: (x3)(x+3).(x3)(x+3).
If you missed this problem, review Example 5.33.

We have seen that some binomials and trinomials result from special products—squaring binomials and multiplying conjugates. If you learn to recognize these kinds of polynomials, you can use the special products patterns to factor them much more quickly.

Factor Perfect Square Trinomials

Some trinomials are perfect squares. They result from multiplying a binomial times itself. We squared a binomial using the Binomial Squares pattern in a previous chapter.

In open parentheses 3x plus 4 close parentheses squared, 3x is a and 4 is b. Writing it as a squared plus 2ab plus b squared, we get open parentheses 3x close parentheses squared plus 2 times 3x times 4 plus 4 squared. This is equal to 9 x squared plus 24x plus 16.

The trinomial 9x2+24x+169x2+24x+16 is called a perfect square trinomial. It is the square of the binomial 3x+4.3x+4.

In this chapter, you will start with a perfect square trinomial and factor it into its prime factors.

You could factor this trinomial using the methods described in the last section, since it is of the form ax2+bx+c.ax2+bx+c. But if you recognize that the first and last terms are squares and the trinomial fits the perfect square trinomials pattern, you will save yourself a lot of work.

Here is the pattern—the reverse of the binomial squares pattern.

Perfect Square Trinomials Pattern

If a and b are real numbers

a2+2ab+b2=(a+b)2a22ab+b2=(ab)2a2+2ab+b2=(a+b)2a22ab+b2=(ab)2

To make use of this pattern, you have to recognize that a given trinomial fits it. Check first to see if the leading coefficient is a perfect square, a2.a2. Next check that the last term is a perfect square, b2.b2. Then check the middle term—is it the product, 2ab?2ab? If everything checks, you can easily write the factors.

Example 6.23

How to Factor Perfect Square Trinomials

Factor: 9x2+12x+4.9x2+12x+4.

Try It 6.45

Factor: 4x2+12x+9.4x2+12x+9.

Try It 6.46

Factor: 9y2+24y+16.9y2+24y+16.

The sign of the middle term determines which pattern we will use. When the middle term is negative, we use the pattern a22ab+b2,a22ab+b2, which factors to (ab)2.(ab)2.

The steps are summarized here.

How To

Factor perfect square trinomials.

Step 1.Does the trinomial fit the pattern?a2+2ab+b2a22ab+b2 Is the first term a perfect square?(a)2(a)2 Write it as a square. Is the last term a perfect square?(a)2(b)2(a)2(b)2 Write it as a square. Check the middle term. Is it2ab?(a)22·a·b(b)2(a)22·a·b(b)2 Step 2.Write the square of the binomial.(a+b)2(ab)2 Step 3.Check by multiplying.Step 1.Does the trinomial fit the pattern?a2+2ab+b2a22ab+b2 Is the first term a perfect square?(a)2(a)2 Write it as a square. Is the last term a perfect square?(a)2(b)2(a)2(b)2 Write it as a square. Check the middle term. Is it2ab?(a)22·a·b(b)2(a)22·a·b(b)2 Step 2.Write the square of the binomial.(a+b)2(ab)2 Step 3.Check by multiplying.

We’ll work one now where the middle term is negative.

Example 6.24

Factor: 81y272y+16.81y272y+16.

Try It 6.47

Factor: 64y280y+25.64y280y+25.

Try It 6.48

Factor: 16z272z+81.16z272z+81.

The next example will be a perfect square trinomial with two variables.

Example 6.25

Factor: 36x2+84xy+49y2.36x2+84xy+49y2.

Try It 6.49

Factor: 49x2+84xy+36y2.49x2+84xy+36y2.

Try It 6.50

Factor: 64m2+112mn+49n2.64m2+112mn+49n2.

Remember the first step in factoring is to look for a greatest common factor. Perfect square trinomials may have a GCF in all three terms and it should be factored out first. And, sometimes, once the GCF has been factored, you will recognize a perfect square trinomial.

Example 6.26

Factor: 100x2y80xy+16y.100x2y80xy+16y.

Try It 6.51

Factor: 8x2y24xy+18y.8x2y24xy+18y.

Try It 6.52

Factor: 27p2q+90pq+75q.27p2q+90pq+75q.

Factor Differences of Squares

The other special product you saw in the previous chapter was the Product of Conjugates pattern. You used this to multiply two binomials that were conjugates. Here’s an example:

We have open parentheses 3x minus 4 close parentheses open parentheses 3x plus 4. This is of the form a minus b, a plus b. We rewrite as open parentheses 3x close parentheses squared minus 4 squared. Here, 3x is a and 4 is b. This is equal to 9 x squared minus 16.

A difference of squares factors to a product of conjugates.

Difference of Squares Pattern

If a and b are real numbers,

a squared minus b squared equals a minus b, a plus b. Here, a squared minus b squared is difference of squares and a minus b, a plus b are conjugates.

Remember, “difference” refers to subtraction. So, to use this pattern you must make sure you have a binomial in which two squares are being subtracted.

Example 6.27

How to Factor a Binomial Using the Difference of Squares

Factor: 64y21.64y21.

Try It 6.53

Factor: 121m21.121m21.

Try It 6.54

Factor: 81y21.81y21.

How To

Factor differences of squares.

Step 1.Does the binomial fit the pattern?a2b2Is this a difference?________Are the first and last terms perfect squares?Step 2.Write them as squares.(a)2(b)2Step 3.Write the product of conjugates.(ab)(a+b)Step 4.Check by multiplying.Step 1.Does the binomial fit the pattern?a2b2Is this a difference?________Are the first and last terms perfect squares?Step 2.Write them as squares.(a)2(b)2Step 3.Write the product of conjugates.(ab)(a+b)Step 4.Check by multiplying.

It is important to remember that sums of squares do not factor into a product of binomials. There are no binomial factors that multiply together to get a sum of squares. After removing any GCF, the expression a2+b2a2+b2 is prime!

The next example shows variables in both terms.

Example 6.28

Factor: 144x249y2.144x249y2.

Try It 6.55

Factor: 196m225n2.196m225n2.

Try It 6.56

Factor: 121p29q2.121p29q2.

As always, you should look for a common factor first whenever you have an expression to factor. Sometimes a common factor may “disguise” the difference of squares and you won’t recognize the perfect squares until you factor the GCF.

Also, to completely factor the binomial in the next example, we’ll factor a difference of squares twice!

Example 6.29

Factor: 48x4y2243y2.48x4y2243y2.

Try It 6.57

Factor: 2x4y232y2.2x4y232y2.

Try It 6.58

Factor: 7a4c27b4c2.7a4c27b4c2.

The next example has a polynomial with 4 terms. So far, when this occurred we grouped the terms in twos and factored from there. Here we will notice that the first three terms form a perfect square trinomial.

Example 6.30

Factor: x26x+9y2.x26x+9y2.

Try It 6.59

Factor: x210x+25y2.x210x+25y2.

Try It 6.60

Factor: x2+6x+94y2.x2+6x+94y2.

Factor Sums and Differences of Cubes

There is another special pattern for factoring, one that we did not use when we multiplied polynomials. This is the pattern for the sum and difference of cubes. We will write these formulas first and then check them by multiplication.

a3+b3=(a+b)(a2ab+b2)a3b3=(ab)(a2+ab+b2)a3+b3=(a+b)(a2ab+b2)a3b3=(ab)(a2+ab+b2)

We’ll check the first pattern and leave the second to you.

.
Distribute. .
Multiply. .
Combine like terms. .

Sum and Difference of Cubes Pattern

a3+b3=(a+b)(a2ab+b2)a3b3=(ab)(a2+ab+b2)a3+b3=(a+b)(a2ab+b2)a3b3=(ab)(a2+ab+b2)

The two patterns look very similar, don’t they? But notice the signs in the factors. The sign of the binomial factor matches the sign in the original binomial. And the sign of the middle term of the trinomial factor is the opposite of the sign in the original binomial. If you recognize the pattern of the signs, it may help you memorize the patterns.

a cubed plus b cubed is open parentheses a plus b close parentheses open parentheses a squared minus ab plus b squared close parentheses. a cubed minus b cubed is open parentheses a minus close parentheses open parentheses a squared plus ab plus b squared close parentheses. In both cases, the sign of the first term on the right side of the equation is the same as the sign on the left side of the equation and the sign of the second term is the opposite of the sign on the left side.

The trinomial factor in the sum and difference of cubes pattern cannot be factored.

It will be very helpful if you learn to recognize the cubes of the integers from 1 to 10, just like you have learned to recognize squares. We have listed the cubes of the integers from 1 to 10 in Table 6.1.

n 1 2 3 4 5 6 7 8 9 10
n3n3 1 8 27 64 125 216 343 512 729 1000
Table 6.1

Example 6.31

How to Factor the Sum or Difference of Cubes

Factor: x3+64.x3+64.

Try It 6.61

Factor: x3+27.x3+27.

Try It 6.62

Factor: y3+8.y3+8.

How To

Factor the sum or difference of cubes.

  1. Step 1. Does the binomial fit the sum or difference of cubes pattern?
    Is it a sum or difference?
    Are the first and last terms perfect cubes?
  2. Step 2. Write them as cubes.
  3. Step 3. Use either the sum or difference of cubes pattern.
  4. Step 4. Simplify inside the parentheses.
  5. Step 5. Check by multiplying the factors.

Example 6.32

Factor: 27u3125v3.27u3125v3.

Try It 6.63

Factor: 8x327y3.8x327y3.

Try It 6.64

Factor: 1000m3125n3.1000m3125n3.

In the next example, we first factor out the GCF. Then we can recognize the sum of cubes.

Example 6.33

Factor: 6x3y+48y4.6x3y+48y4.

Try It 6.65

Factor: 500p3+4q3.500p3+4q3.

Try It 6.66

Factor: 432c3+686d3.432c3+686d3.

The first term in the next example is a binomial cubed.

Example 6.34

Factor: (x+5)364x3.(x+5)364x3.

Try It 6.67

Factor: (y+1)327y3.(y+1)327y3.

Try It 6.68

Factor: (n+3)3125n3.(n+3)3125n3.

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with factoring special products.

Section 6.3 Exercises

Practice Makes Perfect

Factor Perfect Square Trinomials

In the following exercises, factor completely using the perfect square trinomials pattern.

159.

16y2+24y+916y2+24y+9

160.

25v2+20v+425v2+20v+4

161.

36s2+84s+4936s2+84s+49

162.

49s2+154s+12149s2+154s+121

163.

100x220x+1100x220x+1

164.

64z216z+164z216z+1

165.

25n2120n+14425n2120n+144

166.

4p252p+1694p252p+169

167.

49x2+28xy+4y249x2+28xy+4y2

168.

25r2+60rs+36s225r2+60rs+36s2

169.

100y220y+1100y220y+1

170.

64m216m+164m216m+1

171.

10jk2+80jk+160j10jk2+80jk+160j

172.

64x2y96xy+36y64x2y96xy+36y

173.

75u430u3v+3u2v275u430u3v+3u2v2

174.

90p4+300p3q+250p2q290p4+300p3q+250p2q2

Factor Differences of Squares

In the following exercises, factor completely using the difference of squares pattern, if possible.

175.

25v2125v21

176.

169q21169q21

177.

449x2449x2

178.

12125s212125s2

179.

6p2q254p26p2q254p2

180.

98r372r98r372r

181.

24p2+5424p2+54

182.

20b2+14020b2+140

183.

121x2144y2121x2144y2

184.

49x281y249x281y2

185.

169c236d2169c236d2

186.

36p249q236p249q2

187.

16z4116z41

188.

m4n4m4n4

189.

162a4b232b2162a4b232b2

190.

48m4n2243n248m4n2243n2

191.

x216x+64y2x216x+64y2

192.

p2+14p+49q2p2+14p+49q2

193.

a2+6a+99b2a2+6a+99b2

194.

m26m+916n2m26m+916n2

Factor Sums and Differences of Cubes

In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

195.

x3+125x3+125

196.

n6+512n6+512

197.

z627z627

198.

v3216v3216

199.

8343t38343t3

200.

12527w312527w3

201.

8y3125z38y3125z3

202.

27x364y327x364y3

203.

216a3+125b3216a3+125b3

204.

27y3+8z327y3+8z3

205.

7k3+567k3+56

206.

6x348y36x348y3

207.

2x216x2y32x216x2y3

208.

−2x3y216y5−2x3y216y5

209.

(x+3)3+8x3(x+3)3+8x3

210.

(x+4)327x3(x+4)327x3

211.

(y5)364y3(y5)364y3

212.

(y5)3+125y3(y5)3+125y3

Mixed Practice

In the following exercises, factor completely.

213.

64a22564a225

214.

121x2144121x2144

215.

27q2327q23

216.

4p21004p2100

217.

16x272x+8116x272x+81

218.

36y2+12y+136y2+12y+1

219.

8p2+28p2+2

220.

81x2+16981x2+169

221.

1258y31258y3

222.

27u3+100027u3+1000

223.

45n2+60n+2045n2+60n+20

224.

48q324q2+3q48q324q2+3q

225.

x210x+25y2x210x+25y2

226.

x2+12x+36y2x2+12x+36y2

227.

(x+1)3+8x3(x+1)3+8x3

228.

(y3)364y3(y3)364y3

Writing Exercises

229.

Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?

230.

How do you recognize the binomial squares pattern?

231.

Explain why n2+25(n+5)2.n2+25(n+5)2. Use algebra, words, or pictures.

232.

Maribel factored y230y+81y230y+81 as (y9)2.(y9)2. Was she right or wrong? How do you know?

Self Check

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

This table has 4 columns 3 rows and a header row. The header row labels each column I can, confidently, with some help and no, I don’t get it. The first column has the following statements: factor perfect square trinomials, factor differences of squares, factor sums and differences of cubes. The remaining columns are blank.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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