Exercises: Special Products (5.4)

Exercises: Square a Binomial Using the Binomial Squares Pattern

Instructions: For questions 1-20, square each binomial using the Binomial Squares Pattern.

1. [latex]{\left(w+4\right)}^{2}[/latex]
2. [latex]{\left(q+12\right)}^{2}[/latex]
Solution

[latex]{q}^{2}+24q+144[/latex]

3. [latex]{\left(y+\frac{1}{4}\right)}^{2}[/latex]
4. [latex]{\left(x+\frac{2}{3}\right)}^{2}[/latex]
Solution

[latex]{x}^{2}+\frac{4}{3}x+\frac{4}{9}[/latex]

5. [latex]{\left(b-7\right)}^{2}[/latex]
6. [latex]{\left(y-6\right)}^{2}[/latex]
Solution

[latex]{y}^{2}-12y+36[/latex]

7. [latex]{\left(m-15\right)}^{2}[/latex]
8. [latex]{\left(p-13\right)}^{2}[/latex]
Solution

[latex]{p}^{2}-26p+169[/latex]

9. [latex]{\left(3d+1\right)}^{2}[/latex]
10. [latex]{\left(4a+10\right)}^{2}[/latex]
Solution

[latex]16{a}^{2}+80a+100[/latex]

11. [latex]{\left(2q+\frac{1}{3}\right)}^{2}[/latex]
12. [latex]{\left(3z+\frac{1}{5}\right)}^{2}[/latex]
Solution

[latex]9{z}^{2}+\frac{6}{5}z+\frac{1}{25}[/latex]

13. [latex]{\left(3x-y\right)}^{2}[/latex]
14. [latex]{\left(2y-3z\right)}^{2}[/latex]
Solution

[latex]4{y}^{2}-12yz+9{z}^{2}[/latex]

15. [latex]{\left(\frac{1}{5}x-\frac{1}{7}y\right)}^{2}[/latex]
16. [latex]{\left(\frac{1}{8}x-\frac{1}{9}y\right)}^{2}[/latex]
Solution

[latex]\frac{1}{64}{x}^{2}-\frac{1}{36}xy+\frac{1}{81}{y}^{2}[/latex]

17. [latex]{\left(3{x}^{2}+2\right)}^{2}[/latex]
18. [latex]{\left(5{u}^{2}+9\right)}^{2}[/latex]
Solution

[latex]25{u}^{4}+90{u}^{2}+81[/latex]

19. [latex]{\left(4{y}^{3}-2\right)}^{2}[/latex]
20. [latex]{\left(8{p}^{3}-3\right)}^{2}[/latex]
Solution

[latex]64{p}^{6}-48{p}^{3}+9[/latex]

Exercises: Multiply Conjugates Using the Product of Conjugates Pattern

Instructions: For questions 21-44, multiply each pair of conjugates using the Product of Conjugates Pattern.

21. [latex]\left(m-7\right)\left(m+7\right)[/latex]
22. [latex]\left(c-5\right)\left(c+5\right)[/latex]
Solution

[latex]{c}^{2}-25[/latex]

23. [latex]\left(x+\frac{3}{4}\right)\left(x-\frac{3}{4}\right)[/latex]
24. [latex]\left(b+\frac{6}{7}\right)\left(b-\frac{6}{7}\right)[/latex]
Solution

[latex]{b}^{2}-\frac{36}{49}[/latex]

25. [latex]\left(5k+6\right)\left(5k-6\right)[/latex]
26. [latex]\left(8j+4\right)\left(8j-4\right)[/latex]
Solution

[latex]64{j}^{2}-16[/latex]

27. [latex]\left(11k+4\right)\left(11k-4\right)[/latex]
28. [latex]\left(9c+5\right)\left(9c-5\right)[/latex]
Solution

[latex]81{c}^{2}-25[/latex]

29. [latex]\left(11-b\right)\left(11+b\right)[/latex]
30. [latex]\left(13-q\right)\left(13+q\right)[/latex]
Solution

[latex]169-{q}^{2}[/latex]

31. [latex]\left(5-3x\right)\left(5+3x\right)[/latex]
32. [latex]\left(4-6y\right)\left(4+6y\right)[/latex]
Solution

[latex]16-36{y}^{2}[/latex]

33. [latex]\left(9c-2d\right)\left(9c+2d\right)[/latex]
34. [latex]\left(7w+10x\right)\left(7w-10x\right)[/latex]
Solution

[latex]49{w}^{2}-100{x}^{2}[/latex]

35. [latex]\left(m+\frac{2}{3}n\right)\left(m-\frac{2}{3}n\right)[/latex]
36. [latex]\left(p+\frac{4}{5}q\right)\left(p-\frac{4}{5}q\right)[/latex]
Solution

[latex]{p}^{2}-\frac{16}{25}{q}^{2}[/latex]

37. [latex]\left(ab-4\right)\left(ab+4\right)[/latex]
38. [latex]\left(xy-9\right)\left(xy+9\right)[/latex]
Solution

[latex]{x}^{2}{y}^{2}-81[/latex]

39. [latex]\left(uv-\frac{3}{5}\right)\left(uv+\frac{3}{5}\right)[/latex]
40. [latex]\left(rs-\frac{2}{7}\right)\left(rs+\frac{2}{7}\right)[/latex]
Solution

[latex]{r}^{2}{s}^{2}-\frac{4}{49}[/latex]

41. [latex]\left(2{x}^{2}-3{y}^{4}\right)\left(2{x}^{2}+3{y}^{4}\right)[/latex]
42. [latex]\left(6{m}^{3}-4{n}^{5}\right)\left(6{m}^{3}+4{n}^{5}\right)[/latex]
Solution

[latex]36{m}^{6}-16{n}^{10}[/latex]

43. [latex]\left(12{p}^{3}-11{q}^{2}\right)\left(12{p}^{3}+11{q}^{2}\right)[/latex]
44. [latex]\left(15{m}^{2}-8{n}^{4}\right)\left(15{m}^{2}+8{n}^{4}\right)[/latex]
Solution

[latex]225{m}^{4}-64{n}^{8}[/latex]

Exercises: Recognize and Use the Appropriate Special Product Pattern

Instructions: For questions 45-48, find each product.

45.

a. [latex]\left(p-3\right)\left(p+3\right)[/latex]
b. [latex]{\left(t-9\right)}^{2}[/latex]
c. [latex]{\left(m+n\right)}^{2}[/latex]
d. [latex]\left(2x+y\right)\left(x-2y\right)[/latex]

46.

a. [latex]{\left(2r+12\right)}^{2}[/latex]
b. [latex]\left(3p+8\right)\left(3p-8\right)[/latex]
c. [latex]\left(7a+b\right)\left(a-7b\right)[/latex]
d. [latex]{\left(k-6\right)}^{2}[/latex]

Solution

a. [latex]4{r}^{2}+48r+144[/latex]
b. [latex]9{p}^{2}-64[/latex]
c. [latex]7{a}^{2}-48ab-7{b}^{2}[/latex]
d. [latex]{k}^{2}-12k+36[/latex]

47.

a. [latex]{\left({a}^{5}-7b\right)}^{2}[/latex]
b. [latex]\left({x}^{2}+8y\right)\left(8x-{y}^{2}\right)[/latex]
c. [latex]\left({r}^{6}+{s}^{6}\right)\left({r}^{6}-{s}^{6}\right)[/latex]
d. [latex]{\left({y}^{4}+2z\right)}^{2}[/latex]

48.

a. [latex]\left({x}^{5}+{y}^{5}\right)\left({x}^{5}-{y}^{5}\right)[/latex]
b. [latex]{\left({m}^{3}-8n\right)}^{2}[/latex]
c. [latex]{\left(9p+8q\right)}^{2}[/latex]
d. [latex]\left({r}^{2}-{s}^{3}\right)\left({r}^{3}+{s}^{2}\right)[/latex]

Solution

a. [latex]{x}^{10}-{y}^{10}[/latex]
b. [latex]{m}^{6}-16{m}^{3}n+64{n}^{2}[/latex]
c. [latex]81{p}^{2}+144pq+64{q}^{2}[/latex]
d. [latex]{r}^{5}+{r}^{2}{s}^{2}-{r}^{3}{s}^{3}-{s}^{5}[/latex]

Exercises: Everyday Math

Instructions: For questions 49-50, answer the given everyday math word problems.

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

a. Multiply [latex]\left(50-3\right)\left(50+3\right)[/latex] by using the product of conjugates pattern, [latex]\left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}[/latex].
b. Multiply [latex]47\cdot 53[/latex] without using a calculator.
c. Which way is easier for you? Why?

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

a. Multiply [latex]{\left(60+5\right)}^{2}[/latex] by using the binomial squares pattern, [latex]{\left(a+b\right)}^{2}={a}^{2}+2ab+{b}^{2}[/latex].
b. Square [latex]65[/latex] without using a calculator.
c. Which way is easier for you? Why?

Solution

a. [latex]4\text{,}225[/latex]
b. [latex]4\text{,}225[/latex]
c. Answers will vary.

Exercises: Writing Exercises

Instructions: For questions 51-54, answer the given writing exercises.
51. How do you decide which pattern to use?
52. Why does [latex]{\left(a+b\right)}^{2}[/latex] result in a trinomial, but [latex]\left(a-b\right)\left(a+b\right)[/latex] result in a binomial?
Solution

Answers will vary.

53. Marta did the following work on her homework paper:

[latex]\begin{array}{c}{\left(3-y\right)}^{2}\\ {3}^{2}-{y}^{2}\\ 9-{y}^{2}\end{array}[/latex]

Explain what is wrong with Marta’s work.

54. Use the order of operations to show that [latex]{\left(3+5\right)}^{2}[/latex] is [latex]64[/latex], and then use that numerical example to explain why [latex]{\left(a+b\right)}^{2}\ne {a}^{2}+{b}^{2}[/latex].
Solution

Answers will vary.

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Fanshawe Pre-Health Sciences Mathematics 1 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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