6.4E: Exercises
- Page ID
- 30254
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \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
Square a Binomial Using the Binomial Squares Pattern
In the following exercises, square each binomial using the Binomial Squares Pattern.
\((w+4)^2\)
\((q+12)^2\)
- Answer
-
\(q^2+24q+144\)
\((y+14)^2\)
\((x+\frac{2}{3})^2\)
- Answer
-
\(x^2+\frac{4}{3}x+\frac{4}{9}\)
\((b−7)^2\)
\((y−6)^2\)
- Answer
-
\(y^2−12y+36\)
\((m−15)^2\)
\((p−13)^2\)
- Answer
-
\(p^2−26p+169\)
\((3d+1)^2\)
\((4a+10)^2\)
- Answer
-
\(16a^2+80a+100\)
\((2q+13)^2\)
\((3z+15)^2\)
- Answer
-
\(9z^2+65z+125\)
\((3x−y)^2\)
\((2y−3z)^2\)
- Answer
-
\(4y^2−12yz+9z^2\)
\((15x−17y)^2\)
\((18x−19y)^2\)
- Answer
-
\(164x^2−136xy+181y^2\)
\((3x2+2)^2\)
\((5u^2+9)^2\)
- Answer
-
\(25u^4+90u^2+81\)
\((4y^3−2)^2\)
\((8p^3−3)^2\)
- Answer
-
\(64p^6−48p^3+9\)
In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.
\((m−7)(m+7)\)
\((c−5)(c+5)\)
- Answer
-
\(c^2−25\)
\((x+34)(x−34)\)
\((b+\frac{6}{7})(b−\frac{6}{7})\)
- Answer
-
\(b^2−\frac{36}{49}\)
\((5k+6)(5k−6)\)
\((8j+4)(8j−4)\)
- Answer
-
\(64j^2−16\)
\((11k+4)(11k−4)\)
\((9c+5)(9c−5)\)
- Answer
-
\(81c^2−25\)
\((11−b)(11+b)\)
\((13−q)(13+q)\)
- Answer
-
\(169−q^2\)
\((5−3x)(5+3x)\)
\((4−6y)(4+6y)\)
- Answer
-
\(16−36y^2\)
\((9c−2d)(9c+2d)\)
\((7w+10x)(7w−10x)\)
- Answer
-
\(49w^2−100x^2\)
\((m+\frac{2}{3}n)(m−\frac{2}{3}n)\)
\((p+\frac{4}{5}q)(p−\frac{4}{5}q)\)
- Answer
-
\(p^2−\frac{16}{25}q^2\)
\((ab−4)(ab+4)\)
\((xy−9)(xy+9)\)
- Answer
-
\(x^{2}y^2−81\)
\((uv−\frac{3}{5})(uv+\frac{3}{5})\)
\((rs−\frac{2}{7})(rs+\frac{2}{7})\)
- Answer
-
\(r^{2}s^2−\frac{4}{49}\)
\((2x^2−3y^4)(2x^2+3y^4)\)
\((6m^3−4n^5)(6m^3+4n^5)\)
- Answer
-
\(36m^6−16n^{10}\)
\((12p^3−11q^2)(12p^3+11q^2)\)
\((15m^2−8n^4)(15m^2+8n^4)\)
- Answer
-
\(225m^4−64n^8\)
Recognize and Use the Appropriate Special Product Pattern
In the following exercises, find each product.
a. \((p−3)(p+3)\)
b. \((t−9)^2\)
c. \((m+n)^2\)
d. \((2x+y)(x−2y)\)
a. \((2r+12)^2\)
b. \((3p+8)(3p−8)\)
c. \((7a+b)(a−7b)\)
d. \((k−6)^2\)
- Answer
-
a. \(4r^2+48r+144\)
b. \(9p^2−64\)
c. \(7a^2−48ab−7b^2\)
d. \(k^2−12k+36\)
a. \((a^5−7b)^2\)
b. \((x^2+8y)(8x−y^2)\)
c. \((r^6+s^6)(r^6−s^6)\)
d. \((y^4+2z)^2\)
a. \((x^5+y^5)(x^5−y^5)\)
b. \((m^3−8n)^2\)
c. \((9p+8q)^2\)
d. \((r^2−s^3)(r^3+s^2)\)
- Answer
-
a. \(x^{10}−y^{10}\)
b. \(m^6−16m^{3}n+64n^2\)
c. \(81p^2+144pq+64q^2\)
d. \(r^5+r^{2}s^2−r^{3}s^3−s^5\)
Everyday Math
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
- Multiply (50−3)(50+3) by using the product of conjugates pattern, \((a−b)(a+b)=a^2−b^2\)
- Multiply 47·53 without using a calculator.
- Which way is easier for you? Why?
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.
- Multiply \((60+5)^2\) by using the binomial squares pattern, \((a+b)^2=a^2+2ab+b^2\)
- Square 65 without using a calculator.
- Which way is easier for you? Why?
- Answer
-
- 4,225
- 4,225
- Answers will vary.
Writing Exercises
How do you decide which pattern to use?
Why does \((a+b)^2\) result in a trinomial, but (a−b)(a+b) result in a binomial?
- Answer
-
Answers will vary.
Marta did the following work on her homework paper:
\[\begin{array}{c} {(3−y)^2}\\ {3^2−y^2}\\ {9−y^2}\\ \nonumber \end{array}\]
Explain what is wrong with Marta’s work.
Use the order of operations to show that \((3+5)^2\) is 64, and then use that numerical example to explain why \((a+b)^2 \ne a^2+b^2\)
- Answer
-
Answers will vary.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ 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?