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6.4 Special Products

Squaring a binomial and multiplying conjugates by pattern, and deciding which shortcut fits before grinding through term-by-term multiplication.

Elementary Algebra 2e, 6.4 Special Products

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52 exercises

6.4 Exercise 1

Before you get started, take this readiness quiz.

Simplify: ⓐ 929^{2} ⓑ (−9)2\left(−9\right)^{2} ⓒ −92.− 9^{2} .
If you missed this problem, review the reference.

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6.4 Exercise 3

Multiply: (x+9)2.\left(x + 9\right)^{2} .

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6.4 Exercise 4

Multiply: (y+11)2.\left(y + 11\right)^{2} .

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6.4 Exercise 6

Multiply: (x−9)2.\left(x - 9\right)^{2} .

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6.4 Exercise 7

Multiply: (p−13)2.\left(p - 13\right)^{2} .

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6.4 Exercise 9

Multiply: (6x+3)2.\left(6 x + 3\right)^{2} .

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6.4 Exercise 10

Multiply: (4x+9)2.\left(4 x + 9\right)^{2} .

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6.4 Exercise 12

Multiply: (2c−d)2.\left(2 c - d\right)^{2} .

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6.4 Exercise 13

Multiply: (4x−5y)2.\left(4 x - 5 y\right)^{2} .

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6.4 Exercise 15

Multiply: (2x2+1)2.\left(2 x^{2} + 1\right)^{2} .

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6.4 Exercise 16

Multiply: (3y3+2)2.\left(3 y^{3} + 2\right)^{2} .

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6.4 Exercise 18

Multiply: (x−5)(x+5).\left(x - 5\right) \left(x + 5\right) .

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6.4 Exercise 19

Multiply: (w−3)(w+3).\left(w - 3\right) \left(w + 3\right) .

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6.4 Exercise 21

Multiply: (6x+5)(6x−5).\left(6 x + 5\right) \left(6 x - 5\right) .

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6.4 Exercise 22

Multiply: (2x+7)(2x−7).\left(2 x + 7\right) \left(2 x - 7\right) .

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6.4 Exercise 24

Multiply: (7+4x)(7−4x).\left(7 + 4 x\right) \left(7 - 4 x\right) .

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6.4 Exercise 25

Multiply: (9−2y)(9+2y).\left(9 - 2 y\right) \left(9 + 2 y\right) .

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6.4 Exercise 27

Find the product: (4p−7q)(4p+7q).\left(4 p - 7 q\right) \left(4 p + 7 q\right) .

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6.4 Exercise 28

Find the product: (3x−y)(3x+y).\left(3 x - y\right) \left(3 x + y\right) .

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6.4 Exercise 30

Find the product: (xy−6)(xy+6).\left(x y - 6\right) \left(x y + 6\right) .

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6.4 Exercise 31

Find the product: (ab−9)(ab+9).\left(a b - 9\right) \left(a b + 9\right) .

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6.4 Exercise 33

Find the product: (3x2−4y3)(3x2+4y3).\left(3 x^{2} - 4 y^{3}\right) \left(3 x^{2} + 4 y^{3}\right) .

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6.4 Exercise 34

Find the product: (2m2−5n3)(2m2+5n3).\left(2 m^{2} - 5 n^{3}\right) \left(2 m^{2} + 5 n^{3}\right) .

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6.4 Exercise 36

Choose the appropriate pattern and use it to find the product:

ⓐ (9b−2)(2b+9)\left(9 b - 2\right) \left(2 b + 9\right) ⓑ (9p−4)2\left(9 p - 4\right)^{2} ⓒ (7y+1)2\left(7 y + 1\right)^{2} ⓓ (4r−3)(4r+3)\left(4 r - 3\right) \left(4 r + 3\right)

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6.4 Exercise 37

Choose the appropriate pattern and use it to find the product:

ⓐ (6x+7)2\left(6 x + 7\right)^{2} ⓑ (3x−4)(3x+4)\left(3 x - 4\right) \left(3 x + 4\right) ⓒ (2x−5)(5x−2)\left(2 x - 5\right) \left(5 x - 2\right) ⓓ (6n−1)2\left(6 n - 1\right)^{2}

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6.4 Exercise 39

Square each binomial using the Binomial Squares Pattern.

(q+12)2\left(q + 12\right)^{2}

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6.4 Exercise 41

Square each binomial using the Binomial Squares Pattern.

(x+23)2\left(x + \frac{2}{3}\right)^{2}

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6.4 Exercise 43

Square each binomial using the Binomial Squares Pattern.

(y−6)2\left(y - 6\right)^{2}

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6.4 Exercise 45

Square each binomial using the Binomial Squares Pattern.

(p−13)2\left(p - 13\right)^{2}

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6.4 Exercise 47

Square each binomial using the Binomial Squares Pattern.

(4a+10)2\left(4 a + 10\right)^{2}

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6.4 Exercise 49

Square each binomial using the Binomial Squares Pattern.

(3z+15)2\left(3 z + \frac{1}{5}\right)^{2}

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6.4 Exercise 51

Square each binomial using the Binomial Squares Pattern.

(2y−3z)2\left(2 y - 3 z\right)^{2}

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6.4 Exercise 53

Square each binomial using the Binomial Squares Pattern.

(18x−19y)2\left(\frac{1}{8} x - \frac{1}{9} y\right)^{2}

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6.4 Exercise 55

Square each binomial using the Binomial Squares Pattern.

(5u2+9)2\left(5 u^{2} + 9\right)^{2}

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6.4 Exercise 57

Square each binomial using the Binomial Squares Pattern.

(8p3−3)2\left(8 p^{3} - 3\right)^{2}

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6.4 Exercise 59

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(c−5)(c+5)\left(c - 5\right) \left(c + 5\right)

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6.4 Exercise 61

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(b+67)(b−67)\left(b + \frac{6}{7}\right) \left(b - \frac{6}{7}\right)

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6.4 Exercise 63

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(8j+4)(8j−4)\left(8 j + 4\right) \left(8 j - 4\right)

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6.4 Exercise 65

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(9c+5)(9c−5)\left(9 c + 5\right) \left(9 c - 5\right)

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6.4 Exercise 67

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(13−q)(13+q)\left(13 - q\right) \left(13 + q\right)

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6.4 Exercise 69

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(4−6y)(4+6y)\left(4 - 6 y\right) \left(4 + 6 y\right)

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6.4 Exercise 71

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(7w+10x)(7w−10x)\left(7 w + 10 x\right) \left(7 w - 10 x\right)

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6.4 Exercise 73

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(p+45q)(p−45q)\left(p + \frac{4}{5} q\right) \left(p - \frac{4}{5} q\right)

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6.4 Exercise 75

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(xy−9)(xy+9)\left(x y - 9\right) \left(x y + 9\right)

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6.4 Exercise 77

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(rs−27)(rs+27)\left(r s - \frac{2}{7}\right) \left(r s + \frac{2}{7}\right)

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6.4 Exercise 79

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(6m3−4n5)(6m3+4n5)\left(6 m^{3} - 4 n^{5}\right) \left(6 m^{3} + 4 n^{5}\right)

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6.4 Exercise 81

Multiply each pair of conjugates using the Product of Conjugates Pattern.

(15m2−8n4)(15m2+8n4)\left(15 m^{2} - 8 n^{4}\right) \left(15 m^{2} + 8 n^{4}\right)

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6.4 Exercise 83

Find each product.

ⓐ (2r+12)2\left(2 r + 12\right)^{2} ⓑ (3p+8)(3p−8)\left(3 p + 8\right) \left(3 p - 8\right) ⓒ (7a+b)(a−7b)\left(7 a + b\right) \left(a - 7 b\right) ⓓ (k−6)2\left(k - 6\right)^{2}

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6.4 Exercise 85

Find each product.

ⓐ (x5+y5)(x5−y5)\left(x^{5} + y^{5}\right) \left(x^{5} - y^{5}\right) ⓑ (m3−8n)2\left(m^{3} - 8 n\right)^{2} ⓒ (9p+8q)2\left(9 p + 8 q\right)^{2} ⓓ (r2−s3)(r3+s2)\left(r^{2} - s^{3}\right) \left(r^{3} + s^{2}\right)

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6.4 Exercise 87

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+560 + 5.

  1. ⓐ Multiply (60+5)2\left(60 + 5\right)^{2} by using the binomial squares pattern, (a+b)2=a2+2ab+b2\left(a + b\right)^{2} = a^{2} + 2 a b + b^{2}.
  2. ⓑ Square 65 without using a calculator.
  3. ⓒ Which way is easier for you? Why?
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6.4 Exercise 89

Why does (a+b)2\left(a + b\right)^{2} result in a trinomial, but (a−b)(a+b)\left(a - b\right) \left(a + b\right) result in a binomial?

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6.4 Exercise 91

Use the order of operations to show that (3+5)2\left(3 + 5\right)^{2} is 64, and then use that numerical example to explain why (a+b)2≠a2+b2\left(a + b\right)^{2} \neq a^{2} + b^{2}.

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Adapted from Elementary Algebra 2e © 2020 Rice University. Licensed under CC BY 4.0. Source: Elementary Algebra 2e. Imported at revision 77a933c.