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Mathematics LibreTexts

6.4E: Exercises

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    30427
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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

    \(q^2+24q+144\)

    Exercise 3

    \((y+14)^2\)

    Exercise 4

    \((x+\frac{2}{3})^2\)

    Answer

    \(x^2+\frac{4}{3}x+\frac{4}{9}\)

    Exercise 5

    \((b−7)^2\)

    Exercise 6

    \((y−6)^2\)

    Answer

    \(y^2−12y+36\)

    Exercise 7

    \((m−15)^2\)

    Exercise 8

    \((p−13)^2\)

    Answer

    \(p^2−26p+169\)

    Exercise 9

    \((3d+1)^2\)

    Exercise 10

    \((4a+10)^2\)

    Answer

    \(16a^2+80a+100\)

    Exercise 11

    \((2q+13)^2\)

    Exercise 12

    \((3z+15)^2\)

    Answer

    \(9z^2+65z+125\)

    Exercise 13

    \((3x−y)^2\)

    Exercise 14

    \((2y−3z)^2\)

    Answer

    \(4y^2−12yz+9z^2\)

    Exercise 15

    \((15x−17y)^2\)

    Exercise 16

    \((18x−19y)^2\)

    Answer

    \(164x^2−136xy+181y^2\)

    Exercise 17

    \((3x2+2)^2\)

    Exercise 18

    \((5u^2+9)^2\)

    Answer

    \(25u^4+90u^2+81\)

    Exercise 19

    \((4y^3−2)^2\)

    Exercise 20

    \((8p^3−3)^2\)

    Answer

    \(64p^6−48p^3+9\)

    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

    \((m−7)(m+7)\)

    Exercise 22

    \((c−5)(c+5)\)

    Answer

    \(c^2−25\)

    Exercise 23

    \((x+34)(x−34)\)

    Exercise 24

    \((b+\frac{6}{7})(b−\frac{6}{7})\)

    Answer

    \(b^2−\frac{36}{49}\)

    Exercise 25

    \((5k+6)(5k−6)\)

    Exercise 26

    \((8j+4)(8j−4)\)

    Answer

    \(64j^2−16\)

    Exercise 27

    \((11k+4)(11k−4)\)

    Exercise 28

    \((9c+5)(9c−5)\)

    Answer

    \(81c^2−25\)

    Exercise 29

    \((11−b)(11+b)\)

    Exercise 30

    \((13−q)(13+q)\)

    Answer

    \(169−q^2\)

    Exercise 31

    \((5−3x)(5+3x)\)

    Exercise 32

    \((4−6y)(4+6y)\)

    Answer

    \(16−36y^2\)

    Exercise 33

    \((9c−2d)(9c+2d)\)

    Exercise 34

    \((7w+10x)(7w−10x)\)

    Answer

    \(49w^2−100x^2\)

    Exercise 35

    \((m+\frac{2}{3}n)(m−\frac{2}{3}n)\)

    Exercise 36

    \((p+\frac{4}{5}q)(p−\frac{4}{5}q)\)

    Answer

    \(p^2−\frac{16}{25}q^2\)

    Exercise 37

    \((ab−4)(ab+4)\)

    Exercise 38

    \((xy−9)(xy+9)\)

    Answer

    \(x^{2}y^2−81\)

    Exercise 39

    \((uv−\frac{3}{5})(uv+\frac{3}{5})\)

    Exercise 40

    \((rs−\frac{2}{7})(rs+\frac{2}{7})\)

    Answer

    \(r^{2}s^2−\frac{4}{49}\)

    Exercise 41

    \((2x^2−3y^4)(2x^2+3y^4)\)

    Exercise 42

    \((6m^3−4n^5)(6m^3+4n^5)\)

    Answer

    \(36m^6−16n^{10}\)

    Exercise 43

    \((12p^3−11q^2)(12p^3+11q^2)\)

    Exercise 44

    \((15m^2−8n^4)(15m^2+8n^4)\)

    Answer

    \(225m^4−64n^8\)

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

    In the following exercises, find each product.

    Exercise 45

    a. \((p−3)(p+3)\)

    b. \((t−9)^2\)

    c. \((m+n)^2\)

    d. \((2x+y)(x−2y)\)

    Exercise 46

    a. \((2r+12)^2\)

    b. \((3p+8)(3p−8)\)

    c. \((7a+b)(a−7b)\)

    d. \((k−6)^2\)

    Answer

    a. \(4r^2+48r+144\)

    b. \(9p^2−64\)

    c. \(7a^2−48ab−7b^2\)

    d. \(k^2−12k+36\)

    Exercise 47

    a. \((a^5−7b)^2\)

    b. \((x^2+8y)(8x−y^2)\)

    c. \((r^6+s^6)(r^6−s^6)\)

    d. \((y^4+2z)^2\)

    Exercise 48

    a. \((x^5+y^5)(x^5−y^5)\)

    b. \((m^3−8n)^2\)

    c. \((9p+8q)^2\)

    d. \((r^2−s^3)(r^3+s^2)\)

    Answer

    a. \(x^{10}−y^{10}\)

    b. \(m^6−16m^{3}n+64n^2\)

    c. \(81p^2+144pq+64q^2\)

    d. \(r^5+r^{2}s^2−r^{3}s^3−s^5\)

    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, \((a−b)(a+b)=a^2−b^2\)
    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, \((a+b)^2=a^2+2ab+b^2\)
    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:

    \[\begin{array}{c} {(3−y)^2}\\ {3^2−y^2}\\ {9−y^2}\\ \nonumber \end{array}\]

    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 \((a+b)^2 \ne a^2+b^2\)

    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?