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Mathematics LibreTexts

Chapter 6 Review Exercises

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    Chapter Review Exercises

    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.

    \(12a^2b^3,\space 15ab^2\)

    Answer

    \(3ab^2\)

    \(12m^2n^3,42m^5n^3\)

    \(15y^3,\space 21y^2,\space 30y\)

    Answer

    \(3y\)

    \(45x^3y^2,\space 15x^4y,\space 10x^5y^3\)

    Factor the Greatest Common Factor from a Polynomial

    In the following exercises, factor the greatest common factor from each polynomial.

    \(35y+84\)

    Answer

    \(7(5y+12)\)

    \(6y^2+12y−6\)

    \(18x^3−15x\)

    Answer

    \(3x(6x^2−5)\)

    \(15m^4+6m^2n\)

    \(4x^3−12x^2+16x\)

    Answer

    \(4x(x^2−3x+4)\)

    \(−3x+24\)

    \(−3x^3+27x^2−12x\)

    Answer

    \(−3x(x^2−9x+4)\)

    \(3x(x−1)+5(x−1)\)

    Factor by Grouping

    In the following exercises, factor by grouping.

    \(ax−ay+bx−by\)

    Answer

    \((a+b)(x−y)\)

    \(x^2y−xy^2+2x−2y\)

    \(x^2+7x−3x−21\)

    Answer

    \((x−3)(x+7)\)

    \(4x^2−16x+3x−12\)

    \(m^3+m^2+m+1\)

    Answer

    \((m^2+1)(m+1)\)

    \(5x−5y−y+x\)

    Factor Trinomials

    Factor Trinomials of the Form \(x^2+bx+c\)

    In the following exercises, factor each trinomial of the form \(x^2+bx+c\).

    \(a^2+14a+33\)

    Answer

    \((a+3)(a+11)\)

    \(k^2−16k+60\)

    \(m^2+3m−54\)

    Answer

    \((m+9)(m−6)\)

    \(x^2−3x−10\)

    In the following examples, factor each trinomial of the form \(x^2+bxy+cy^2\).

    \(x^2+12xy+35y^2\)

    Answer

    \((x+5y)(x+7y)\)

    \(r^2+3rs−28s^2\)

    \(a^2+4ab−21b^2\)

    Answer

    \((a+7b)(a−3b)\)

    \(p^2−5pq−36q^2\)

    \(m^2−5mn+30n^2\)

    Answer

    Prime

    Factor Trinomials of the Form ax2+bx+cax2+bx+c Using Trial and Error

    In the following exercises, factor completely using trial and error.

    \(x^3+5x^2−24x\)

    \(3y^3−21y^2+30y\)

    Answer

    \(3y(y−5)(y−2)\)

    \(5x^4+10x^3−75x^2\)

    \(5y^2+14y+9\)

    Answer

    \((5y+9)(y+1)\)

    \(8x^2+25x+3\)

    \(10y^2−53y−11\)

    Answer

    \((5y+1)(2y−11)\)

    \(6p^2−19pq+10q^2\)

    \(−81a^2+153a+18\)

    Answer

    \(−9(9a−1)(a+2)\)

    Factor Trinomials of the Form ax2+bx+cax2+bx+c using the ‘ac’ Method

    In the following exercises, factor.

    \(2x^2+9x+4\)

    \(18a^2−9a+1\)

    Answer

    \((3a−1)(6a−1)\)

    \(15p^2+2p−8\)

    \(15x^2+6x−2\)

    Answer

    \((3x−1)(5x+2)\)

    \(8a^2+32a+24\)

    \(3x^2+3x−36\)

    Answer

    \(3(x+4)(x−3)\)

    \(48y^2+12y−36\)

    \(18a^2−57a−21\)

    Answer

    \(3(2a−7)(3a+1)\)

    \(3n^4−12n^3−96n^2\)

    Factor using substitution

    In the following exercises, factor using substitution.

    \(x^4−13x^2−30\)

    Answer

    \((x^2−15)(x^2+2)\)

    \((x−3)^2−5(x−3)−36\)

    Factor Special Products

    Factor Perfect Square Trinomials

    In the following exercises, factor completely using the perfect square trinomials pattern.

    \(25x^2+30x+9\)

    Answer

    \((5x+3)^2\)

    \(36a^2−84ab+49b^2\)

    \(40x^2+360x+810\)

    Answer

    \(10(2x+9)^2\)

    \(5k^3−70k^2+245k\)

    \(75u^4−30u^3v+3u^2v^2\)

    Answer

    \(3u^2(5u−v)^2\)

    Factor Differences of Squares

    In the following exercises, factor completely using the difference of squares pattern, if possible.

    \(81r^2−25\)

    \(169m^2−n^2\)

    Answer

    \((13m+n)(13m−n)\)

    \(25p^2−1\)

    \(9−121y^2\)

    Answer

    \((3+11y)(3−11y)\)

    \(20x^2−125\)

    \(169n^3−n\)

    Answer

    \(n(13n+1)(13n−1)\)

    \(6p^2q^2−54p^2\)

    \(24p^2+54\)

    Answer

    \(6(4p^2+9)\)

    \(49x^2−81y^2\)

    \(16z^4−1\)

    Answer

    \((2z−1)(2z+1)(4z^2+1)\)

    \(48m^4n^2−243n^2\)

    \(a^2+6a+9−9b^2\)

    Answer

    \((a+3−3b)(a+3+3b)\)

    \(x^2−16x+64−y^2\)

    Factor Sums and Differences of Cubes

    In the following exercises, factor completely using the sums and differences of cubes pattern, if possible.

    \(a^3−125\)

    Answer

    \((a−5)(a^2+5a+25)\)

    \(b^3−216\)

    \(2m^3+54\)

    Answer

    \(2(m+3)(m^2−3m+9)\)

    \(81m^3+3\)

    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

    \((4n−7m)^2\)

    \(6a^2−25a−9\)

    \(5u^4−45u^2\)

    Answer

    \(5u^2(u+3)(u−3)\)

    \(n^4−81\)

    \(64j^2+225\)

    Answer

    prime

    \(5x^2+5x−60\)

    \(b^3−64\)

    Answer

    \((b−4)(b^2+4b+16)\)

    \(m^3+125\)

    \(2b^2−2bc+5cb−5c^2\)

    Answer

    \((2b+5c)(b−c)\)

    \(48x^5y^2−243xy^2\)

    \(5q^2−15q−90\)

    Answer

    \(5(q+3)(q−6) \)

    \(4u^5v+4u^2v^3\)

    \(10m^4−6250\)

    Answer

    \(10(m−5)(m+5)(m^2+25)\)

    \(60x^2y−75xy+30y\)

    \(16x^2−24xy+9y^2−64\)

    Answer

    \((4x−3y+8)(4x−3y−8)\)

    Polynomial Equations

    Use the Zero Product Property

    In the following exercises, solve.

    \((a−3)(a+7)=0\)

    \((5b+1)(6b+1)=0\)

    Answer

    \(b=−\frac{1}{5},\space b=−\frac{1}{6}\)

    \(6m(12m−5)=0\)

    \((2x−1)^2=0\)

    Answer

    \(x=\frac{1}{2}\)

    \(3m(2m−5)(m+6)=0\)

    Solve Quadratic Equations by Factoring

    In the following exercises, solve.

    \(x^2+9x+20=0\)

    Answer

    \(x=−4,\space x=−5\)

    \(y^2−y−72=0\)

    \(2p^2−11p=40\)

    Answer

    \(p=−\frac{5}{2},p=8\)

    \(q^3+3q^2+2q=0\)

    \(144m^2−25=0\)

    Answer

    \(m=\frac{5}{12},\space m=−\frac{5}{12}\)

    \(4n^2=36\)

    \((x+6)(x−3)=−8\)

    Answer

    \(x=2,\space x=−5\)

    \((3x−2)(x+4)=12\)

    \(16p^3=24p^2+9p\)

    Answer

    \(p=0,\space p=\frac{3}{4}\)

    \(2y^3+2y^2=12y\)

    Solve Equations with Polynomial Functions

    In the following exercises, solve.

    For the function, \(f(x)=x^2+11x+20\), ⓐ find when \(f(x)=−8\) ⓑ Use this information to find two points that lie on the graph of the function.

    Answer

    ⓐ \(x=−7\) or \\(x=−4\)
    ⓑ \((−7,−8)\) \((−4,−8)\)

    For the function, \(f(x)=9x^2−18x+5\), ⓐ find when \(f(x)=−3\) ⓑ Use this information to find two points that lie on the graph of the function.

    In each function, find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒ the y-intercept of the graph of the function.

    \(f(x)=64x^2−49\)

    Answer

    ⓐ \(x=\frac{7}{8}\) or \(x=−\frac{7}{8}\)
    ⓑ \((\frac{7}{8},0),\space (−\frac{7}{8},0)\) ⓒ \((0,−49)\)

    \(f(x)=6x^2−13x−5\)

    Solve Applications Modeled by Quadratic Equations

    In the following exercises, solve.

    The product of two consecutive numbers is 399. Find the numbers.

    Answer

    The numbers are \(−21\) and \(−19\) or 19 and 21.

    The area of a rectangular shaped patio 432 square feet. The length of the patio is 6 feet more than its width. Find the length and width.

    A ladder leans against the wall of a building. The length of the ladder is 9 feet longer than the distance of the bottom of the ladder from the building. The distance of the top of the ladder reaches up the side of the building is 7 feet longer than the distance of the bottom of the ladder from the building. Find the lengths of all three sides of the triangle formed by the ladder leaning against the building.

    Answer

    The lengths are 8, 15, and 17 ft.

    Shruti is going to throw a ball from the top of a cliff. When she throws the ball from 80 feet above the ground, the function \(h(t)=−16t^2+64t+80\) models the height, h, of the ball above the ground as a function of time, t. Find: ⓐthe zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 80 feet above the ground. ⓒ the height the ball will be at \(t=2\) seconds which is when the ball will be at its highest point.

    Chapter Practice Test

    In the following exercises, factor completely.

    \(80a^2+120a^3\)

    Answer

    \(40a^2(2+3a)\)

    \(5m(m−1)+3(m−1)\)

    \(x^2+13x+36\)

    Answer

    \((x+7)(x+6)\)

    \(p^2+pq−12q^2\)

    \(xy−8y+7x−56\)

    Answer

    \((x−8)(y+7)\)

    \(40r^2+810\)

    \(9s^2−12s+4\)

    Answer

    \((3s−2)^2\)

    \(6x^2−11x−10\)

    \(3x^2−75y^2\)

    Answer

    \(3(x+5y)(x−5y)\)

    \(6u^2+3u−18\)

    \(x^3+125\)

    Answer

    \((x+5)(x^2−5x+25)\)

    \(32x^5y^2−162xy^2\)

    \(6x^4−19x^2+15\)

    Answer

    \((3x^2−5)(2x^2−3)\)

    \(3x^3−36x^2+108x\)

    In the following exercises, solve

    \(5a^2+26a=24\)

    Answer

    \(a=\frac{4}{5},\space a=−6\)

    The product of two consecutive integers is 156. Find the integers.

    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.

    Answer

    The width is 12 inches and the length is 14 inches.

    Jing is going to throw a ball from the balcony of her condo. When she throws the ball from 80 feet above the ground, the function \(h(t)=−16t^2+64t+80\) models the height, h, of the ball above the ground as a function of time, t. Find: ⓐthe zeros of this function which tells us when the ball will hit the ground. ⓑ the time(s) the ball will be 128 feet above the ground. ⓒ the height the ball will be at \(t=4\) seconds.

    For the function, \(f(x)=x^2−7x+5\), ⓐ find when \(f(x)=−7\) ⓑ Use this information to find two points that lie on the graph of the function.

    Answer

    ⓐ \(x=3\) or \(x=4\) ⓑ \((3,−7)\) \((4,−7)\)

    For the function \(f(x)=25x^2−81\), find: ⓐ the zeros of the function ⓑ the x-intercepts of the graph of the function ⓒthe y-intercept of the graph of the function.

    Glossary

    degree of the polynomial equation
    The degree of the polynomial equation is the degree of the polynomial.
    polynomial equation
    A polynomial equation is an equation that contains a polynomial expression.
    quadratic equation
    Polynomial equations of degree two are called quadratic equations.
    zero of the function
    A value of xx where the function is 0, is called a zero of the function.
    Zero Product Property
    The Zero Product Property says that if the product of two quantities is zero, then at least one of the quantities is zero.
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