# 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. ${\left(w+4\right)}^{2}$

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

${q}^{2}+24q+144$

3. ${\left(y+\frac{1}{4}\right)}^{2}$

4. ${\left(x+\frac{2}{3}\right)}^{2}$
Solution

${x}^{2}+\frac{4}{3}x+\frac{4}{9}$

5. ${\left(b-7\right)}^{2}$

6. ${\left(y-6\right)}^{2}$
Solution

${y}^{2}-12y+36$

7. ${\left(m-15\right)}^{2}$

8. ${\left(p-13\right)}^{2}$
Solution

${p}^{2}-26p+169$

9. ${\left(3d+1\right)}^{2}$

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

$16{a}^{2}+80a+100$

11. ${\left(2q+\frac{1}{3}\right)}^{2}$

12. ${\left(3z+\frac{1}{5}\right)}^{2}$
Solution

$9{z}^{2}+\frac{6}{5}z+\frac{1}{25}$

13. ${\left(3x-y\right)}^{2}$

14. ${\left(2y-3z\right)}^{2}$
Solution

$4{y}^{2}-12yz+9{z}^{2}$

15. ${\left(\frac{1}{5}x-\frac{1}{7}y\right)}^{2}$

16. ${\left(\frac{1}{8}x-\frac{1}{9}y\right)}^{2}$
Solution

$\frac{1}{64}{x}^{2}-\frac{1}{36}xy+\frac{1}{81}{y}^{2}$

17. ${\left(3{x}^{2}+2\right)}^{2}$

18. ${\left(5{u}^{2}+9\right)}^{2}$
Solution

$25{u}^{4}+90{u}^{2}+81$

19. ${\left(4{y}^{3}-2\right)}^{2}$

20. ${\left(8{p}^{3}-3\right)}^{2}$
Solution

$64{p}^{6}-48{p}^{3}+9$

## 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. $\left(m-7\right)\left(m+7\right)$

22. $\left(c-5\right)\left(c+5\right)$
Solution

${c}^{2}-25$

23. $\left(x+\frac{3}{4}\right)\left(x-\frac{3}{4}\right)$

24. $\left(b+\frac{6}{7}\right)\left(b-\frac{6}{7}\right)$
Solution

${b}^{2}-\frac{36}{49}$

25. $\left(5k+6\right)\left(5k-6\right)$

26. $\left(8j+4\right)\left(8j-4\right)$
Solution

$64{j}^{2}-16$

27. $\left(11k+4\right)\left(11k-4\right)$

28. $\left(9c+5\right)\left(9c-5\right)$
Solution

$81{c}^{2}-25$

29. $\left(11-b\right)\left(11+b\right)$

30. $\left(13-q\right)\left(13+q\right)$
Solution

$169-{q}^{2}$

31. $\left(5-3x\right)\left(5+3x\right)$

32. $\left(4-6y\right)\left(4+6y\right)$
Solution

$16-36{y}^{2}$

33. $\left(9c-2d\right)\left(9c+2d\right)$

34. $\left(7w+10x\right)\left(7w-10x\right)$
Solution

$49{w}^{2}-100{x}^{2}$

35. $\left(m+\frac{2}{3}n\right)\left(m-\frac{2}{3}n\right)$

36. $\left(p+\frac{4}{5}q\right)\left(p-\frac{4}{5}q\right)$
Solution

${p}^{2}-\frac{16}{25}{q}^{2}$

37. $\left(ab-4\right)\left(ab+4\right)$

38. $\left(xy-9\right)\left(xy+9\right)$
Solution

${x}^{2}{y}^{2}-81$

39. $\left(uv-\frac{3}{5}\right)\left(uv+\frac{3}{5}\right)$

40. $\left(rs-\frac{2}{7}\right)\left(rs+\frac{2}{7}\right)$
Solution

${r}^{2}{s}^{2}-\frac{4}{49}$

41. $\left(2{x}^{2}-3{y}^{4}\right)\left(2{x}^{2}+3{y}^{4}\right)$

42. $\left(6{m}^{3}-4{n}^{5}\right)\left(6{m}^{3}+4{n}^{5}\right)$
Solution

$36{m}^{6}-16{n}^{10}$

43. $\left(12{p}^{3}-11{q}^{2}\right)\left(12{p}^{3}+11{q}^{2}\right)$

44. $\left(15{m}^{2}-8{n}^{4}\right)\left(15{m}^{2}+8{n}^{4}\right)$
Solution

$225{m}^{4}-64{n}^{8}$

## Exercises: Recognize and Use the Appropriate Special Product Pattern

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

45.

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

46.

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

Solution

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

47.

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

48.

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

Solution

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

## 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 $47$ times $53$. Think of $47$ as $50-3$ and $53$ as $50+3$.

a. Multiply $\left(50-3\right)\left(50+3\right)$ by using the product of conjugates pattern, $\left(a-b\right)\left(a+b\right)={a}^{2}-{b}^{2}$.
b. Multiply $47\cdot 53$ 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 $65$. Think of $65$ as $60+5$.

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

Solution

a. $4\text{,}225$
b. $4\text{,}225$
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 ${\left(a+b\right)}^{2}$ result in a trinomial, but $\left(a-b\right)\left(a+b\right)$ result in a binomial?
Solution

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

$\begin{array}{c}{\left(3-y\right)}^{2}\\ {3}^{2}-{y}^{2}\\ 9-{y}^{2}\end{array}$

Explain what is wrong with Marta’s work.

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