7.4E: Exercises

Last updated
Save as PDF

Page ID: 30556

Contributed by Lynn Marecek
Professor (Mathematics) at Santa Ana College
Publisher: OpenStax CNX

$ \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } $ $ \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} $$\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$ $\newcommand{\id}{\mathrm{id}}$ $ \newcommand{\Span}{\mathrm{span}}$ $ \newcommand{\kernel}{\mathrm{null}\,}$ $ \newcommand{\range}{\mathrm{range}\,}$ $ \newcommand{\RealPart}{\mathrm{Re}}$ $ \newcommand{\ImaginaryPart}{\mathrm{Im}}$ $ \newcommand{\Argument}{\mathrm{Arg}}$ $ \newcommand{\norm}[1]{\| #1 \|}$ $ \newcommand{\inner}[2]{\langle #1, #2 \rangle}$ $ \newcommand{\Span}{\mathrm{span}}$

Practice Makes Perfect

Factor Perfect Square Trinomials

In the following exercises, factor.

Exercise 1

$16y^2+24y+9$

Answer: $(4y+3)^2$

Exercise 2

$25v^2+20v+4$

Exercise 3

$36s^2+84s+49$

Answer: $(6s+7)^2$

Exercise 4

$49s^2+154s+121$

Exercise 5

$100x^2−20x+1$

Answer: $(10x−1)^2$

Exercise 6

$64z^2−16z+1$

Exercise 7

$25n^2−120n+144$

Answer: $(5n−12)^2$

Exercise 8

$4p^2−52p+169$

Exercise 9

$49x^2−28xy+4y^2$

Answer: $(7x−2y)^2$

Exercise 10

$25r^2−60rs+36s^2$

Exercise 11

$25n^2+25n+4$

Answer: $(5n+4)(5n+1)$

Exercise 12

$100y^2−20y+1$

Exercise 13

$64m^2−16m+1$

Answer: $(8m-1)^2$

Exercise 14

$100x^2−25x+1$

Exercise 15

$10k^2+80k+160$

Answer: $10(k+4)^2$

Exercise 16

$64x^2−96x+36$

Exercise 17

$75u^3−30u^{2}v+3uv^2$

Answer: $3u(5u−v)^2$

Exercise 18

$90p^3+300p^{2}q+250pq^2$

Factor Differences of Squares

In the following exercises, factor.

Exercise 19

$x^2−16$

Answer: $(x−4)(x+4)$

Exercise 20

$n^2−9$

Exercise 21

$25v^2−1$

Answer: $(5v−1)(5v+1)$

Exercise 22

$169q^2−1$

Exercise 23

$121x^2−144y^2$

Answer: $(11x−12y)(11x+12y)$

Exercise 24

$49x^2−81y^2$

Exercise 25

$169c^2−36d^2$

Answer: $(13c−6d)(13c+6d)$

Exercise 26

$36p^2−49q^2$

Exercise 27

$4−49x^2$

Answer: $(2−7x)(2+7x)$

Exercise 28

$121−25s^2$

Exercise 29

$16z^4−1$

Answer: $(2z−1)(2z+1)(4z^2+1)$

Exercise 30

$m^4−n^4$

Exercise 31

$5q^2−45$

Answer: $5(q−3)(q+3)$

Exercise 32

$98r^3−72r$

Exercise 33

$24p^2+54$

Answer: $6(4p^2+9)$

Exercise 34

$20b^2+140$

Factor Sums and Differences of Cubes

In the following exercises, factor.

Exercise 35

$x^3+125$

Answer: $(x+5)(x^2−5x+25)$

Exercise 36

$n^3+512$

Exercise 37

$z^3−27$

Answer: $(z−3)(z^2+3z+9)$

Exercise 38

$v^3−216$

Exercise 39

$8−343t^3$

Answer: $(2−7t)(4+14t+49t^2)$

Exercise 40

$125−27w^3$

Exercise 41

$8y^3−125z^3$

Answer: $(2y−5z)(4y^2+10yz+25z^2)$

Exercise 42

$27x^3−64y^3$

Exercise 43

$7k^3+56$

Answer: $7(k+2)(k^2−2k+4)$

Exercise 44

$6x^3−48y^3$

Exercise 45

$2−16y^3$

Answer: $2(1−2y)(1+2y+4y^2)$

Exercise 46

$−2x^3−16y^3$

Mixed Practice

In the following exercises, factor.

Exercise 47

$64a^2−25$

Answer: $(8a−5)(8a+5)$

Exercise 48

$121x^2−144$

Exercise 49

$27q^2−3$

Answer: $3(3q−1)(3q+1)$

Exercise 50

$4p^2−100$

Exercise 51

$16x^2−72x+81$

Answer: $(4x−9)^2$

Exercise 52

$36y^2+12y+1$

Exercise 53

$8p^2+2$

Answer: $2(4p^2+1)^2$

Exercise 54

$81x^2+169$

Exercise 55

$125−8y^3$

Answer: $(5−2y)(25+10y+4y^2)$

Exercise 56

$27u^3+1000$

Exercise 57

$45n^2+60n+20$

Answer: $5(3n+2)^2$

Exercise 58

$48q^3−24q^2+3q$

Everyday Math

Exercise 59

Landscaping Sue and Alan are planning to put a $15$ foot square swimming pool in their backyard. They will surround the pool with a tiled deck, the same width on all sides. If the width of the deck is $w$, the total area of the pool and deck is given by the trinomial $4w^2+60w+225$.

Answer: $(2w+15)^2$

Exercise 60

Home repair The height a twelve foot ladder can reach up the side of a building if the ladder’s base is $b$ feet from the building is the square root of the binomial $144−b^2$.

Writing Exercises

Exercise 61

Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?

Answer: Answers may vary.

Exercise 62

How do you recognize the binomial squares pattern?

Exercise 63

Explain why $n^2+25 \ne (n+5)^2$.

Answer: Answers may vary.

Exercise 64

Maribel factored $y^2−30y+81$ as (y−9)^2. How do you know that this is incorrect?

Self Check

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

$This table has the following statements all to be preceded by “I can…”. The first row is “factor perfect square trinomials”. The second row is “factor differences of squares”. The third row is “factor sums and differences of cubes”. In the columns beside these statements are the headers, “confidently”, “with some help”, and “no-I don’t get it!”.$

b. 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?