Chapter 7 Review Exercises
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapter 7 Review Exercises
7.1 Greatest Common Factor and Factor by Grouping
Find the Greatest Common Factor of Two or More Expressions
In the following exercises, find the greatest common factor.
42, 60
- Answer
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6
450, 420
90, 150, 105
- Answer
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15
60, 294, 630
Factor the Greatest Common Factor from a Polynomial
In the following exercises, factor the greatest common factor from each polynomial.
\(24x−42\)
- Answer
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\(6(4x−7)\)
\(35y+84\)
\(15m^4+6m^{2}n\)
- Answer
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\(3m^2(5m2+2n)\)
\(24pt^4+16t^7\)
Factor by Grouping
In the following exercises, factor by grouping.
\(ax−ay+bx−by\)
- Answer
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\((a+b)(x−y)\)
\(x^{2}y−xy^2+2x−2y\)
\(x^2+7x−3x−21\)
- Answer
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\((x−3)(x+7)\)
\(4x^2−16x+3x−12\)
\(m^3+m^2+m+1\)
- Answer
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\((m^2+1)(m+1)\)
\(5x−5y−y+x\)
7.2 Factor Trinomials of the form \(x^2+bx+c\)
Factor Trinomials of the Form \(x^2+bx+c\)
In the following exercises, factor each trinomial of the form \(x^2+bx+c\)
\(u^2+17u+72\)
- Answer
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\((u+8)(u+9)\)
\(a^2+14a+33\)
\(k^2−16k+60\)
- Answer
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\((k−6)(k−10)\)
\(r^2−11r+28\)
\(y^2+6y−7\)
- Answer
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\((y+7)(y−1)\)
\(m^2+3m−54\)
\(s^2−2s−8\)
- Answer
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\((s−4)(s+2)\)
\(x^2−3x−10\)
Factor Trinomials of the Form \(x^2+bxy+cy^2\)
In the following examples, factor each trinomial of the form \(x^2+bxy+cy^2\)
\(x^2+12xy+35y^2\)
- Answer
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\((x+5y)(x+7y)\)
\(u^2+14uv+48v^2\)
\(a^2+4ab−21b^2\)
- Answer
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\((a+7b)(a−3b)\)
\(p^2−5pq−36q^2\)
7.3 Factoring Trinomials of the form \(ax^2+bx+c\)
Recognize a Preliminary Strategy to Factor Polynomials Completely
In the following exercises, identify the best method to use to factor each polynomial.
\(y^2−17y+42\)
- Answer
-
Undo FOIL
\(12r^2+32r+5\)
\(8a^3+72a\)
- Answer
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Factor the GCF
\(4m−mn−3n+12\)
Factor Trinomials of the Form \(ax^2+bx+c\) with a GCF
In the following exercises, factor completely.
\(6x^2+42x+60\)
- Answer
-
\(6(x+2)(x+5)\)
\(8a^2+32a+24\)
\(3n^4−12n^3−96n^2\)
- Answer
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\(3n^{2}(n−8)(n+4)\)
\(5y^4+25y^2−70y\)
Factor Trinomials Using the “ac” Method
In the following exercises, factor.
\(2x^2+9x+4\)
- Answer
-
\((x+4)(2x+1)\)
\(3y^2+17y+10\)
\(18a^2−9a+1\)
- Answer
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\((3a−1)(6a−1)\)
\(8u^2−14u+3\)
\(15p^2+2p−8\)
- Answer
-
\((5p+4)(3p−2)\)
\(15x^2+6x−2\)
\(40s^2−s−6\)
- Answer
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\((5s−2)(8s+3)\)
\(20n^2−7n−3\)
Factor Trinomials with a GCF Using the “ac” Method
In the following exercises, factor.
\(3x^2+3x−36\)
- Answer
-
\(3(x+4)(x−3)\)
\(4x^2+4x−8\)
\(60y^2−85y−25\)
- Answer
-
\(5(4y+1)(3y−5)\)
\(18a^2−57a−21\)
7.4 Factoring Special Products
Factor Perfect Square Trinomials
In the following exercises, factor.
\(25x^2+30x+9\)
- Answer
-
\((5x+3)^2\)
\(16y^2+72y+81\)
\(36a^2−84ab+49b^2\)
- Answer
-
\((6a−7b)^2\)
\(64r^2−176rs+121s^2\)
\(40x^2+360x+810\)
- Answer
-
\(10(2x+9)^2\)
\(75u^2+180u+108\)
\(2y^3−16y^2+32y\)
- Answer
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\(2y(y−4)^2\)
\(5k^3−70k^2+245k\)
In the following exercises, factor.
\(81r^2−25\)
- Answer
-
\((9r−5)(9r+5)\)
\(49a^2−144\)
\(169m^2−n^2\)
- Answer
-
\((13m+n)(13m−n)\)
\(64x^2−y^2\)
\(25p^2−1\)
- Answer
-
\((5p−1)(5p+1)\)
\(1−16s^2\)
\(9−121y^2\)
- Answer
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\((3+11y)(3−11y)\)
\(100k^2−81\)
\(20x^2−125\)
- Answer
-
\(5(2x−5)(2x+5)\)
\(18y^2−98\)
\(49u^3−9u\)
- Answer
-
\(u(7u+3)(7u−3)\)
\(169n^3−n\)
Factor Sums and Differences of Cubes
In the following exercises, factor.
\(a^3−125\)
- Answer
-
\((a−5)(a^2+5a+25)\)
\(b^3−216\)
\(2m^3+54\)
- Answer
-
\(2(m+3)(m^2−3m+9)\)
\(81x^3+3\)
7.5 General Strategy for Factoring Polynomials
Recognize and Use the Appropriate Method to Factor a Polynomial Completely
In the following exercises, factor completely.
\(24x^3+44x^2\)
- Answer
-
\(4x^{2}(6x+11)\)
\(24a^4−9a^3\)
\(16n^2−56mn+49m^2\)
- Answer
-
\((4n−7m)^2\)
\(6a^2−25a−9\)
\(5r^2+22r−48\)
- Answer
-
(r+6)(5r−8)
\(5u^4−45u^2\)
\(n^4−81\)
- Answer
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\((n^2+9)(n+3)(n−3)\)
\(64j^2+225\)
\(5x^2+5x−60\)
- Answer
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\(5(x−3)(x+4)\)
\(b^3−64\)
\(m^3+125\)
- Answer
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\((m+5)(m^2−5m+25)\)
\(2b^2−2bc+5cb−5c^2\)
7.6 Quadratic Equations
Use the Zero Product Property
In the following exercises, solve.
\((a−3)(a+7)=0\)
- Answer
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\(a=3\), \(a=−7\)
\((b−3)(b+10)=0\)
\(3m(2m−5)(m+6)=0\)
- Answer
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\(m=0\), \(m=−6\), \(m=\frac{5}{2}\)
\(7n(3n+8)(n−5)=0\)
Solve Quadratic Equations by Factoring
In the following exercises, solve.
\(x^2+9x+20=0\)
- Answer
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\(x=−4\), \(x=−5\)
\(y^2−y−72=0\)
\(2p^2−11p=40\)
- Answer
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\(p=−\frac{5}{2}\), p=8
\(q^3+3q^2+2q=0\)
\(144m^2−25=0\)
- Answer
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\(m=\frac{5}{12}\), \(m=−\frac{5}{12}\)
\(4n^2=36\)
Solve Applications Modeled by Quadratic Equations
In the following exercises, solve.
The product of two consecutive numbers is 462.
- Answer
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−21,−22
21, 22
The area of a rectangular shaped patio 400 square feet. The length of the patio is 99 feet more than its width. Find the length and width.
Practice Test
In the following exercises, find the Greatest Common Factor in each expression.
\(14y−42\)
- Answer
-
\(7(y−6)\)
\(−6x^2−30x\)
\(80a^2+120a^3\)
- Answer
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\(40a^{2}(2+3a)\)
\(5m(m−1)+3(m−1)\)
In the following exercises, factor completely.
\(x^2+13x+36\)
- Answer
-
\((x+7)(x+6)\)
\(p^2+pq−12q^2\)
\(3a^3−6a^2−72a\)
- Answer
-
\(3a(a+4)(a-6)\)
\(s^2−25s+84\)
\(5n^2+30n+45\)
- Answer
-
\(5(n+3)^2\)
\(64y^2−49\)
\(xy−8y+7x−56\)
- Answer
-
\((x−8)(y+7)\)
\(40r^2+810\)
\(9s^2−12s+4\)
- Answer
-
\((3s−2)^2\)
\(n^2+12n+36\)
\(100−a^2\)
- Answer
-
\((10−a)(10+a)\)
\(6x^2−11x−10\)
\(3x^2−75y^2\)
- Answer
-
\(3(x+5y)(x−5y)\)
\(c^3−1000d^3\)
\(ab−3b−2a+6\)
- Answer
-
\((a−3)(b−2)\)
\(6u^2+3u−18\)
\(8m^2+22m+5\)
- Answer
-
\((4m+1)(2m+5)\)
In the following exercises, solve.
\(x^2+9x+20=0\)
\(y^2=y+132\)
- Answer
-
\(y=−11\), \(y=12\)
\(5a^2+26a=24\)
\(9b^2−9=0\)
- Answer
-
\(b=1\), \(b=−1\)
\(16−m^2=0\)
\(4n^2+19n+21=0\)
- Answer
-
\(n=−\frac{7}{4}\), n=−3
\((x−3)(x+2)=6\)
The product of two consecutive integers is 156.
- Answer
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12 and 13; −13 and −12
The area of a rectangular place mat is 168 square inches. Its length is two inches longer than the width. Find the length and width of the placemat.