6.6E: Exercises

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Dec 3, 2023
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- 6.6: Divide Polynomials
- 6.7: Integer Exponents and Scientific Notation

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Practice Makes Perfect

In the following exercises, divide each polynomial by the monomial.

Exercise 1

$\frac{45 y + 36}{9}$

Exercise 2

$\frac{30 b + 75}{5}$

Answer: $6 b + 15$

Exercise 3

$\frac{8 d^{2} - 4 d}{2}$

Exercise 4

$\frac{42 x^{2} - 14 x}{7}$

Answer: $6 x^{2} - 2 x$

Exercise 5

$(16 y^{2} - 20 y) \div 4 y$

Exercise 6

$(55 w^{2} - 10 w) \div 5 w$

Answer: $11 w - 2$

Exercise 7

$(9 n^{4} + 6 n^{3}) \div 3 n$

Exercise 8

$(8 x^{3} + 6 x^{2}) \div 2 x$

Answer: $4 x^{2} + 3 x$

Exercise 9

$\frac{18 y^{2} - 12 y}{- 6}$

Exercise 10

$\frac{20 b^{2} - 12 b}{- 4}$

Answer: $- 5 b^{2} + 3 b$

Exercise 11

$\frac{35 a^{4} + 65 a^{2}}{- 5}$

Exercise 12

$\frac{51 m^{4} + 72 m^{3}}{- 3}$

Answer: $- 17 m^{4} - 24 m^{3}$

Exercise 13

$\frac{310 y^{4} - 200 y^{3}}{5 y^{2}}$

Exercise 14

$\frac{412 z^{8} - 48 z^{5}}{4 z^{3}}$

Answer: $103 z^{5} - 12 z^{2}$

Exercise 15

$\frac{46 x^{3} + 38 x^{2}}{2 x^{2}}$

Exercise 16

$\frac{51 y^{4} + 42 y^{2}}{3 y^{2}}$

Answer: $17 y^{2} + 14$

Exercise 17

$(24 p^{2} - 33 p) \div (- 3 p)$

Exercise 18

$(35 x^{4} - 21 x) \div (- 7 x)$

Answer: $- 5 x^{3} + 3$

Exercise 19

$(63 m^{4} - 42 m^{3}) \div (- 7 m^{2})$

Exercise 20

$(48 y^{4} - 24 y^{3}) \div (- 8 y^{2})$

Answer: $- 6 y^{2} + 3 y$

Exercise 21

$(63 a^{2} b^{3} + 72 a b^{4}) \div (9 a b)$

Exercise 22

$(45 x^{3} y^{4} + 60 x y^{2}) \div (5 x y)$

Answer: $9 x^{2} y^{3} + 12 y$

Exercise 23

$\frac{52 p^{5} q^{4} + 36 p^{4} q^{3} - 64 p^{3} q^{2}}{4 p^{2} q}$

Exercise 24

$\frac{49 c^{2} d^{2} - 70 c^{3} d^{3} - 35 c^{2} d}{7 c d^{2}}$

Answer: $7 c - 10 c^{2} d - \frac{5 c}{d}$

Exercise 25

$\frac{66 x^{3} y^{2} - 110 x^{2} y^{3} - 44 x^{4} y^{3}}{11 x^{2} y^{2}}$

Exercise 26

$\frac{72 r^{5} s^{2} + 132 r^{4} s^{3} - 96 r^{3} s^{5}}{12 r^{2} s^{2}}$

Answer: $6 r^{3} + 11 r^{2} s - 8 r s^{3}$

Exercise 27

$\frac{4 w^{2} + 2 w - 5}{2 w}$

Exercise 28

$\frac{12 q^{2} + 3 q - 1}{3 q}$

Answer: $4 q + 1 - \frac{1}{3 q}$

Exercise 29

$\frac{10 x^{2} + 5 x - 4}{- 5 x}$

Exercise 30

$\frac{20 y^{2} + 12 y - 1}{- 4 y}$

Answer: $- 5 y - 3 + \frac{1}{4 y}$

Exercise 31

$\frac{36 p^{3} + 18 p^{2} - 12 p}{6 p^{2}}$

Exercise 32

$\frac{63 a^{3} - 108 a^{2} + 99 a}{9 a^{2}}$

Answer: $7 a - 12 + \frac{11}{a}$

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

Exercise 33

$(y^{2} + 7 y + 12) \div (y + 3)$

Exercise 34

$(d^{2} + 8 d + 12) \div (d + 2)$

Answer: $d + 6$

Exercise 35

$(x^{2} - 3 x - 10) \div (x + 2)$

Exercise 36

$(a^{2} - 2 a - 35) \div (a + 5)$

Answer: $a - 7$

Exercise 37

$(t^{2} - 12 t + 36) \div (t - 6)$

Exercise 38

$(x^{2} - 14 x + 49) \div (x - 7)$

Answer: $x - 7$

Exercise 39

$(6 m^{2} - 19 m - 20) \div (m - 4)$

Exercise 40

$(4 x^{2} - 17 x - 15) \div (x - 5)$

Answer: $4 x + 3$

Exercise 41

$(q^{2} + 2 q + 20) \div (q + 6)$

Exercise 42

$(p^{2} + 11 p + 16) \div (p + 8)$

Answer: $p + 3 - \frac{8}{p + 8}$

Exercise 43

$(y^{2} - 3 y - 15) \div (y - 8)$

Exercise 44

$(x^{2} + 2 x - 30) \div (x - 5)$

Answer: $x + 7 + \frac{5}{x - 5}$

Exercise 45

$(3 b^{3} + b^{2} + 2) \div (b + 1)$

Exercise 46

$(2 n^{3} - 10 n + 28) \div (n + 3)$

Answer: $2 n^{2} - 6 n + 8 + \frac{4}{n + 3}$

Exercise 47

$(2 y^{3} - 6 y - 36) \div (y - 3)$

Exercise 48

$(7 q^{3} - 5 q - 2) \div (q - 1)$

Answer: $7 q^{2} + 7 q + 2$

Exercise 49

$(z^{3} + 1) \div (z + 1)$

Exercise 50

$(m^{3} + 1000) \div (m + 10)$

Answer: $m^{2} - 10 m + 100$

Exercise 51

$(a^{3} - 125) \div (a - 5)$

Exercise 52

$(x^{3} - 216) \div (x - 6)$

Answer: $x^{2} + 6 x + 36$

Exercise 53

$(64 x^{3} - 27) \div (4 x - 3)$

Exercise 54

$(125 y^{3} - 64) \div (5 y - 4)$

Answer: $25 y^{2} + 20 x + 16$

Everyday Math

Exercise 55

Average cost Pictures Plus produces digital albums. The company’s average cost (in dollars) to make x albums is given by the expression $\frac{7 x + 500}{x}$

Find the quotient by dividing the numerator by the denominator.
What will the average cost (in dollars) be to produce 20 albums?

Exercise 56

Handshakes At a company meeting, every employee shakes hands with every other employee. The number of handshakes is given by the expression $\frac{n^{2} - n}{2}$ nn represents the number of employees. How many handshakes will there be if there are 10 employees at the meeting?

Answer: 45

Writing Exercises

Exercise 57

James divides $48 y + 6$ by $6$ this way: $\frac{48 y + 6}{6} = 48 y$

Exercise 58

Divide $\frac{10 x^{2} + x - 12}{2 x}$ and explain with words how you get each term of the quotient.

Answer: Answers will vary.

Self Check

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

$This is a table that has three rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “divide a polynomial by a monomial,” and “divide a polynomial by a binomial.” The rest of the cells are blank.$

ⓑ After reviewing this checklist, what will you do to become confident for all goals?