7.4E: Exercises
- Page ID
- 30556
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Practice Makes Perfect
Factor Perfect Square Trinomials
In the following exercises, factor.
\(16y^2+24y+9\)
- Answer
-
\((4y+3)^2\)
\(25v^2+20v+4\)
\(36s^2+84s+49\)
- Answer
-
\((6s+7)^2\)
\(49s^2+154s+121\)
\(100x^2−20x+1\)
- Answer
-
\((10x−1)^2\)
\(64z^2−16z+1\)
\(25n^2−120n+144\)
- Answer
-
\((5n−12)^2\)
\(4p^2−52p+169\)
\(49x^2−28xy+4y^2\)
- Answer
-
\((7x−2y)^2\)
\(25r^2−60rs+36s^2\)
\(25n^2+25n+4\)
- Answer
-
\((5n+4)(5n+1)\)
\(100y^2−20y+1\)
\(64m^2−16m+1\)
- Answer
-
\((8m-1)^2\)
\(100x^2−25x+1\)
\(10k^2+80k+160\)
- Answer
-
\(10(k+4)^2\)
\(64x^2−96x+36\)
\(75u^3−30u^{2}v+3uv^2\)
- Answer
-
\(3u(5u−v)^2\)
\(90p^3+300p^{2}q+250pq^2\)
Factor Differences of Squares
In the following exercises, factor.
\(x^2−16\)
- Answer
-
\((x−4)(x+4)\)
\(n^2−9\)
\(25v^2−1\)
- Answer
-
\((5v−1)(5v+1)\)
\(169q^2−1\)
\(121x^2−144y^2\)
- Answer
-
\((11x−12y)(11x+12y)\)
\(49x^2−81y^2\)
\(169c^2−36d^2\)
- Answer
-
\((13c−6d)(13c+6d)\)
\(36p^2−49q^2\)
\(4−49x^2\)
- Answer
-
\((2−7x)(2+7x)\)
\(121−25s^2\)
\(16z^4−1\)
- Answer
-
\((2z−1)(2z+1)(4z^2+1)\)
\(m^4−n^4\)
\(5q^2−45\)
- Answer
-
\(5(q−3)(q+3)\)
\(98r^3−72r\)
\(24p^2+54\)
- Answer
-
\(6(4p^2+9)\)
\(20b^2+140\)
Factor Sums and Differences of Cubes
In the following exercises, factor.
\(x^3+125\)
- Answer
-
\((x+5)(x^2−5x+25)\)
\(n^3+512\)
\(z^3−27\)
- Answer
-
\((z−3)(z^2+3z+9)\)
\(v^3−216\)
\(8−343t^3\)
- Answer
-
\((2−7t)(4+14t+49t^2)\)
\(125−27w^3\)
\(8y^3−125z^3\)
- Answer
-
\((2y−5z)(4y^2+10yz+25z^2)\)
\(27x^3−64y^3\)
\(7k^3+56\)
- Answer
-
\(7(k+2)(k^2−2k+4)\)
\(6x^3−48y^3\)
\(2−16y^3\)
- Answer
-
\(2(1−2y)(1+2y+4y^2)\)
\(−2x^3−16y^3\)
Mixed Practice
In the following exercises, factor.
\(64a^2−25\)
- Answer
-
\((8a−5)(8a+5)\)
\(121x^2−144\)
\(27q^2−3\)
- Answer
-
\(3(3q−1)(3q+1)\)
\(4p^2−100\)
\(16x^2−72x+81\)
- Answer
-
\((4x−9)^2\)
\(36y^2+12y+1\)
\(8p^2+2\)
- Answer
-
\(2(4p^2+1)\)
\(81x^2+169\)
\(125−8y^3\)
- Answer
-
\((5−2y)(25+10y+4y^2)\)
\(27u^3+1000\)
\(45n^2+60n+20\)
- Answer
-
\(5(3n+2)^2\)
\(48q^3−24q^2+3q\)
Everyday Math
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
-
\((2w+15)^2\)
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
Why was it important to practice using the binomial squares pattern in the chapter on multiplying polynomials?
- Answer
-
Answers may vary.
How do you recognize the binomial squares pattern?
Explain why \(n^2+25 \ne (n+5)^2\).
- Answer
-
Answers may vary.
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.
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?