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6.4E: Exercises

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Practice Makes Perfect

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

Exercise 1

(w+4)2

Exercise 2

(q+12)2

Answer

q2+24q+144

Exercise 3

(y+14)2

Exercise 4

(x+23)2

Answer

x2+43x+49

Exercise 5

(b7)2

Exercise 6

(y6)2

Answer

y212y+36

Exercise 7

(m15)2

Exercise 8

(p13)2

Answer

p226p+169

Exercise 9

(3d+1)2

Exercise 10

(4a+10)2

Answer

16a2+80a+100

Exercise 11

(2q+13)2

Exercise 12

(3z+15)2

Answer

9z2+65z+125

Exercise 13

(3xy)2

Exercise 14

(2y3z)2

Answer

4y212yz+9z2

Exercise 15

(15x17y)2

Exercise 16

(18x19y)2

Answer

164x2136xy+181y2

Exercise 17

(3x2+2)2

Exercise 18

(5u2+9)2

Answer

25u4+90u2+81

Exercise 19

(4y32)2

Exercise 20

(8p33)2

Answer

64p648p3+9

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

Exercise 21

(m7)(m+7)

Exercise 22

(c5)(c+5)

Answer

c225

Exercise 23

(x+34)(x34)

Exercise 24

(b+67)(b67)

Answer

b23649

Exercise 25

(5k+6)(5k6)

Exercise 26

(8j+4)(8j4)

Answer

64j216

Exercise 27

(11k+4)(11k4)

Exercise 28

(9c+5)(9c5)

Answer

81c225

Exercise 29

(11b)(11+b)

Exercise 30

(13q)(13+q)

Answer

169q2

Exercise 31

(53x)(5+3x)

Exercise 32

(46y)(4+6y)

Answer

1636y2

Exercise 33

(9c2d)(9c+2d)

Exercise 34

(7w+10x)(7w10x)

Answer

49w2100x2

Exercise 35

(m+23n)(m23n)

Exercise 36

(p+45q)(p45q)

Answer

p21625q2

Exercise 37

(ab4)(ab+4)

Exercise 38

(xy9)(xy+9)

Answer

x2y281

Exercise 39

(uv35)(uv+35)

Exercise 40

(rs27)(rs+27)

Answer

r2s2449

Exercise 41

(2x23y4)(2x2+3y4)

Exercise 42

(6m34n5)(6m3+4n5)

Answer

36m616n10

Exercise 43

(12p311q2)(12p3+11q2)

Exercise 44

(15m28n4)(15m2+8n4)

Answer

225m464n8

​​​​​​Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

Exercise 45

a. (p3)(p+3)

b. (t9)2

c. (m+n)2

d. (2x+y)(x2y)

Exercise 46

a. (2r+12)2

b. (3p+8)(3p8)

c. (7a+b)(a7b)

d. (k6)2

Answer

a. 4r2+48r+144

b. 9p264

c. 7a248ab7b2

d. k212k+36

Exercise 47

a. (a57b)2

b. (x2+8y)(8xy2)

c. (r6+s6)(r6s6)

d. (y4+2z)2

Exercise 48

a. (x5+y5)(x5y5)

b. (m38n)2

c. (9p+8q)2

d. (r2s3)(r3+s2)

Answer

a. x10y10

b. m616m3n+64n2

c. 81p2+144pq+64q2

d. r5+r2s2r3s3s5

Everyday Math

Exercise 49

Mental math You can use the product of conjugates pattern to multiply numbers without a calculator. Say you need to multiply 47 times 53. Think of 47 as 50−3 and 53 as 50+3

  1. Multiply (50−3)(50+3) by using the product of conjugates pattern, (ab)(a+b)=a2b2
  2. Multiply 47·53 without using a calculator.
  3. Which way is easier for you? Why?
Exercise 50

Mental math You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as 60+5.

  1. Multiply (60+5)2 by using the binomial squares pattern, (a+b)2=a2+2ab+b2
  2. Square 65 without using a calculator.
  3. Which way is easier for you? Why?
Answer
  1. 4,225
  2. 4,225
  3. Answers will vary.

Writing Exercises

Exercise 51

How do you decide which pattern to use?

Exercise 52

Why does (a+b)2 result in a trinomial, but (a−b)(a+b) result in a binomial?

Answer

Answers will vary.

Exercise 53

Marta did the following work on her homework paper:

(3y)232y29y2

Explain what is wrong with Marta’s work.

Exercise 54

Use the order of operations to show that (3+5)2 is 64, and then use that numerical example to explain why (a+b)2a2+b2

Answer

Answers will vary.

Self Check

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

This is a table that has four rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “square a binomial using the binomial squares pattern,” “multiply conjugates using the product of conjugates pattern,” and “recognize and use the appropriate special product pattern.” The rest of the cells are blank.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?


This page titled 6.4E: Exercises is shared under a CC BY license and was authored, remixed, and/or curated by OpenStax.

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