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Mathematics LibreTexts

Chapter 5 Review Exercises

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    Chapter Review Exercises

    Add and Subtract Polynomials

    Determine the Degree of Polynomials

    In the following exercises, determine the type of polynomial.

    1.  \(16x^2−40x−25\)

    2.  \(5m+9\)

    Answer

    binomial

    3.  \(−15\)

    4.  \(y^2+6y^3+9y^4\)

    Answer

    other polynomial

    Add and Subtract Polynomials

    In the following exercises, add or subtract the polynomials.

    5.  \(4p+11p\)

    6.  \(−8y^3−5y^3\)

    Answer

    \(−13y^3\)

    7.  \((4a^2+9a−11)+(6a^2−5a+10)\)

    8.  \((8m^2+12m−5)−(2m^2−7m−1)\)

    Answer

    \(6m^2+19m−4\)

    9.  \((y^2−3y+12)+(5y^2−9)\)

    10.  \((5u^2+8u)−(4u−7)\)

    Answer

    \(5u^2+4u+7\)

    11. Find the sum of \(8q^3−27\) and \(q^2+6q−2\).

    12. Find the difference of \(x^2+6x+8\) and \(x^2−8x+15\).

    Answer

    \(2x^2−2x+23\)

    In the following exercises, simplify.

    13.  \(17mn^2−(−9mn^2)+3mn^2\)

    14.  \(18a−7b−21a\)

    Answer

    \(−7b−3a\)

    15.  \(2pq^2−5p−3q^2\)

    16.  \((6a^2+7)+(2a^2−5a−9)\)

    Answer

    \(8a^2−5a−2\)

    17.  \((3p^2−4p−9)+(5p^2+14)\)

    18.  \((7m^2−2m−5)−(4m^2+m−8)\)

    Answer

    \(−3m+3\)

    19.  \((7b^2−4b+3)−(8b^2−5b−7)\)

    20. Subtract \((8y^2−y+9)\) from \( (11y^2−9y−5) \)

    Answer

    \(3y^2−8y−14\)

    21. Find the difference of \((z^2−4z−12)\) and \((3z^2+2z−11)\)

    22.  \((x^3−x^2y)−(4xy^2−y^3)+(3x^2y−xy^2)\)

    Answer

    \(x^3+2x^2y−4xy^2\)

    23.  \((x^3−2x^2y)−(xy^2−3y^3)−(x^2y−4xy^2)\)

    Evaluate a Polynomial Function for a Given Value of the Variable

    In the following exercises, find the function values for each polynomial function.

    24. For the function \(f(x)=7x^2−3x+5\) find:
    a. \(f(5)\)   b. \(f(−2)\)   c. \(f(0)\)

    Answer

    a. 165   b. 39   c. 5

    25. For the function \(g(x)=15−16x^2\), find:
    a. \(g(−1)\)   b. \(g(0)\)   c. \(g(2)\)

    26. A pair of glasses is dropped off a bridge 640 feet above a river. The polynomial function \(h(t)=−16t^2+640\) gives the height of the glasses t seconds after they were dropped. Find the height of the glasses when \(t=6\).

    Answer

    The height is 64 feet.

    27. A manufacturer of the latest soccer shoes has found that the revenue received from selling the shoes at a cost of \(p\) dollars each is given by the polynomial \(R(p)=−5p^2+360p\). Find the revenue received when \(p=110\) dollars.

    Add and Subtract Polynomial Functions

    In the following exercises, find a. \((f + g)(x)\) b. \((f + g)(3)\) c. \((f − g)(x\) d. \((f − g)(−2)\)

    28.  \(f(x)=2x^2−4x−7\) and \(g(x)=2x^2−x+5\)

    Answer

    a. \((f+g)(x)=4x^2−5x−2\)
    b. \((f+g)(3)=19\)
    c. \((f−g)(x)=−3x−12\)
    d. \((f−g)(−2)=−6\)

    29.  \(f(x)=4x^3−3x^2+x−1\) and \(g(x)=8x^3−1\)

    Properties of Exponents and Scientific Notation

    Simplify Expressions Using the Properties for Exponents

    In the following exercises, simplify each expression using the properties for exponents.

    30.  \(p^3·p^{10}\)

    Answer

    \(p^{13}\)

    31.  \(2·2^6\)

    32.  \(a·a^2·a^3\)

    Answer

    \(a^6\)

    33.  \(x·x^8\)

    34.  \(y^a·y^b\)

    Answer

    \(y^{a+b}\)

    35.  \(\dfrac{2^8}{2^2}\)

    36.  \(\dfrac{a^6}{a}\)

    Answer

    \(a^5\)

    37.  \(\dfrac{n^3}{n^{12}}\)

    38.  \(\dfrac{1}{x^5}\)

    Answer

    \(\dfrac{1}{x^4}\)

    39.  \(3^0\)

    40.  \(y^0\)

    Answer

    \(1\)

    41.  \((14t)^0\)

    42.  \(12a^0−15b^0\)

    Answer

    \(−3\)

    Use the Definition of a Negative Exponent

    In the following exercises, simplify each expression.

    43.  \(6^{−2}\)

    44.  \((−10)^{−3}\)

    Answer

    \(−\dfrac{1}{1000}\)

    45.  \(5·2^{−4}\)

    46.  \((8n)^{−1}\)

    Answer

    \(\dfrac{1}{8n}\)

    47.  \(y^{−5}\)

    48.  \(10^{−3}\)

    Answer

    \(\dfrac{1}{1000}\)

    49.  \(\dfrac{1}{a^{−4}}\)

    50.  \(\dfrac{1}{6^{−2}}\)

    Answer

    \(36\)

    51.  \(−5^{−3}\)

    52.  \( \left(−\dfrac{1}{5}\right)^{−3}\)

    Answer

    \(−\dfrac{1}{25}\)

    53.  \(−(12)^{−3}\)

    54.  \((−5)^{−3}\)

    Answer

    \(−\dfrac{1}{125}\)

    55.  \(\left(\dfrac{5}{9}\right)^{−2}\)

    56.  \(\left(−\dfrac{3}{x}\right)^{−3}\)

    Answer

    \(\dfrac{x^3}{27}\)

    In the following exercises, simplify each expression using the Product Property.

    57.  \((y^4)^3\)

    58.  \((3^2)^5\)

    Answer

    \(3^{10}\)

    59.  \((a^{10})^y\)

    60.  \(x^{−3}·x^9\)

    Answer

    \(x^5\)

    61.  \(r^{−5}·r^{−4}\)

    62.  \((uv^{−3})(u^{−4}v^{−2})\)

    Answer

    \(\dfrac{1}{u^3v^5}\)

    63.  \((m^5)^{−1}\)

    64.  \(p^5·p^{−2}·p^{−4}\)

    Answer

    \(\dfrac{1}{m^5}\)

    In the following exercises, simplify each expression using the Power Property.

    65.  \((k−2)^{−3}\)

    66.  \(\dfrac{q^4}{q^{20}}\)

    Answer

    \(\dfrac{1}{q^{16}}\)

    67.  \(\dfrac{b^8}{b^{−2}}\)

    68.  \(\dfrac{n^{−3}}{n^{−5}}\)

    Answer

    \(n^2\)

    In the following exercises, simplify each expression using the Product to a Power Property.

    69.  \((−5ab)^3\)

    70.  \((−4pq)^0\)

    Answer

    \(1\)

    71.  \((−6x^3)^{−2}\)

    72.  \((3y^{−4})^2\)

    Answer

    \(\dfrac{9}{y^8}\)

    In the following exercises, simplify each expression using the Quotient to a Power Property.

    73.  \(\left(\dfrac{3}{5x}\right)^{−2}\)

    74.  \(\left(\dfrac{3xy^2}{z}\right)^4\)

    Answer

    \(\dfrac{81x^4y^8}{z^4}\)

    75.  \((4p−3q^2)^2\)

    In the following exercises, simplify each expression by applying several properties.

    76.  \((x^2y)^2(3xy^5)^3\)

    Answer

    \(27x^7y^{17}\)

    77.  \((−3a^{−2})^4(2a^4)^2(−6a^2)^3\)

    78.  \(\left(\dfrac{3xy^3}{4x^4y^{−2}}\right)^2\left(\dfrac{6xy^4}{8x^3y^{−2}}\right)^{−1}\)

    Answer

    \(\dfrac{3y^4}{4x^4}\)

    In the following exercises, write each number in scientific notation.

    79.  \(2.568\)

    80.  \(5,300,000\)

    Answer

    \(5.3×10^6\)

    81.  \(0.00814\)

    In the following exercises, convert each number to decimal form.

    82.  \(2.9×10^4\)

    Answer

    \(29,000\)

    83.  \(3.75×10^{−1}\)

    84.  \(9.413×10^{−5}\)

    Answer

    \(0.00009413\)

    In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

    85.  \((3×10^7)(2×10^{−4})\)

    86.  \((1.5×10^{−3})(4.8×10^{−1})\)

    Answer

    \(0.00072\)

    87.  \(\dfrac{6×10^9}{2×10^{−1}}\)

    88.  \(\dfrac{9×10^{−3}}{1×10^{−6}}\)

    Answer

    \(9,000\)

    Multiply Polynomials

    Multiply Monomials

    In the following exercises, multiply the monomials.

    89.  \((−6p^4)(9p)\)

    90.  \(\left(\frac{1}{3}c^2\right)(30c^8)\)

    Answer

    \(10c^{10}\)

    91.  \((8x^2y^5)(7xy^6)\)

    92.  \( \left(\frac{2}{3}m^3n^6\right)\left(\frac{1}{6}m^4n^4\right)\)

    Answer

    \(\dfrac{m^7n^{10}}{9}\)

    Multiply a Polynomial by a Monomial

    In the following exercises, multiply.

    93.  \(7(10−x)\)

    94.  \(a^2(a^2−9a−36)\)

    Answer

    \(a^4−9a^3−36a^2\)

    95.  \(−5y(125y^3−1)\)

    96.  \((4n−5)(2n^3)\)

    Answer

    \(8n^4−10n^3\)

    Multiply a Binomial by a Binomial

    In the following exercises, multiply the binomials using:

    a. the Distributive Property b. the FOIL method c. the Vertical Method.

    97.  \((a+5)(a+2)\)

    98.  \((y−4)(y+12)\)

    Answer

    \(y^2+8y−48\)

    99.  \((3x+1)(2x−7)\)

    100.  \((6p−11)(3p−10)\)

    Answer

    \(18p^2−93p+110\)

    In the following exercises, multiply the binomials. Use any method.

    101.  \((n+8)(n+1)\)

    102.  \((k+6)(k−9)\)

    Answer

    \(k^2−3k−54\)

    103.  \((5u−3)(u+8)\)

    104.  \((2y−9)(5y−7)\)

    Answer

    \(10y^2−59y+63\)

    105.  \((p+4)(p+7)\)

    106.  \((x−8)(x+9)\)

    Answer

    \(x^2+x−72\)

    107.  \((3c+1)(9c−4)\)

    108.  \((10a−1)(3a−3)\)

    Answer

    \(30a^2−33a+3\)

    Multiply a Polynomial by a Polynomial

    In the following exercises, multiply using a. the Distributive Property b. the Vertical Method.

    109.  \((x+1)(x^2−3x−21)\)

    110.  \((5b−2)(3b^2+b−9)\)

    Answer

    \(15b^3−b^2−47b+18\)

    In the following exercises, multiply. Use either method.

    111.  \((m+6)(m^2−7m−30)\)

    112.  \((4y−1)(6y^2−12y+5)\)

    Answer

    \(24y^2−54y^2+32y−5\)

    Multiply Special Products

    In the following exercises, square each binomial using the Binomial Squares Pattern.

    113.  \((2x−y)^2\)

    114.  \((x+\dfrac{3}{4})^2\)

    Answer

    \(x^2+\dfrac{3}{2}x+\dfrac{9}{16}\)

    115.  \((8p^3−3)^2\)

    116.  \((5p+7q)^2\)

    Answer

    \(25p^2+70pq+49q^2\)

    In the following exercises, multiply each pair of conjugates using the Product of Conjugates.

    117.  \((3y+5)(3y−5)\)

    118.  \((6x+y)(6x−y)\)

    Answer

    \(36x^2−y^2\)

    119.  \((a+\dfrac{2}3b)(a−\dfrac{2}{3}b)\)

    120.  \((12x^3−7y^2)(12x^3+7y^2)\)

    Answer

    \(144x^6−49y^4\)

    121.  \((13a^2−8b4)(13a^2+8b^4)\)

    Divide Monomials

    Divide Monomials

    In the following exercises, divide the monomials.

    122.  \(72p^{12}÷8p^3\)

    Answer

    \(9p^9\)

    123.  \(−26a^8÷(2a^2)\)

    124.  \(\dfrac{45y^6}{−15y^{10}}\)

    Answer

    \(−3y^4\)

    125.  \(\dfrac{−30x^8}{−36x^9}\)

    126.  \(\dfrac{28a^9b}{7a^4b^3}\)

    Answer

    \(\dfrac{4a^5}{b^2}\)

    127.  \(\dfrac{11u^6v^3}{55u^2v^8}\)

    128.  \(\dfrac{(5m^9n^3)(8m^3n^2)}{(10mn^4)(m^2n^5)}\)

    Answer

    \(\dfrac{4m^9}{n^4}\)

    129.  \(\dfrac{(42r^2s^4)(54rs^2)}{(6rs^3)(9s)}\)

    Divide a Polynomial by a Monomial

    In the following exercises, divide each polynomial by the monomial

    130.  \((54y^4−24y^3)÷(−6y^2)\)

    Answer

    \(−9y^2+4y\)

    131.  \(\dfrac{63x^3y^2−99x^2y^3−45x^4y^3}{9x^2y^2}\)

    132.  \(\dfrac{12x^2+4x−3}{−4x}\)

    Answer

    \(−3x−1+\dfrac{3}{4x}\)

    Divide Polynomials using Long Division

    In the following exercises, divide each polynomial by the binomial.

    133.  \((4x^2−21x−18)÷(x−6)\)

    134.  \((y^2+2y+18)÷(y+5)\)

    Answer

    \(y−3+\dfrac{33}{q+6}\)

    135.  \((n^3−2n^2−6n+27)÷(n+3)\)

    136.  \((a^3−1)÷(a+1)\)

    Answer

    \(a^2+a+1\)

    Divide Polynomials using Synthetic Division

    In the following exercises, use synthetic Division to find the quotient and remainder.

    137.  \(x^3−3x^2−4x+12\) is divided by \(x+2\)

    138.  \(2x^3−11x^2+11x+12\) is divided by \(x−3\)

    Answer

    \(2x^2−5x−4;\space0\)

    139.  \(x^4+x^2+6x−10\) is divided by \(x+2\)

    Divide Polynomial Functions

    In the following exercises, divide.

    140. For functions \(f(x)=x^2−15x+45\) and \(g(x)=x−9\), find a. \(\left(\dfrac{f}{g}\right)(x)\)
    b. \(\left(\dfrac{f}{g}\right)(−2)\)

    Answer

    a. \(\left(\dfrac{f}{g}\right)(x)=x−6\)
    b. \(\left(\dfrac{f}{g}\right)(−2)=−8\)

    141. For functions \(f(x)=x^3+x^2−7x+2\) and \(g(x)=x−2\), find a. \(\left(\dfrac{f}{g}\right)(x)\)
    b. \(\left(\dfrac{f}{g}\right)(3)\)

    Use the Remainder and Factor Theorem

    In the following exercises, use the Remainder Theorem to find the remainder.

    142.  \(f(x)=x^3−4x−9\) is divided by \(x+2\)

    Answer

    \(−9\)

    143.  \(f(x)=2x^3−6x−24\) divided by \(x−3\)

    In the following exercises, use the Factor Theorem to determine if \(x−c\) is a factor of the polynomial function.

    144. Determine whether \(x−2\) is a factor of \(x^3−7x^2+7x−6\)

    Answer

    no

    145. Determine whether \(x−3\) is a factor of \(x^3−7x^2+11x+3\)

     

    Chapter Practice Test

    1. For the polynomial \(8y^4−3y^2+1\)

    a. Is it a monomial, binomial, or trinomial? b. What is its degree?

    Answer

    a. trinomial b. 4

    2.  \((5a^2+2a−12)(9a^2+8a−4)\)

    3.  \((10x^2−3x+5)−(4x^2−6)\)

    Answer

    \(6x^2−3x+11\)

    4.  \(\left(−\dfrac{3}{4}\right)^3\)

    5.  \(x^{−3}x^4\)

    Answer

    \(x\)

    6.  \(5^65^8\)

    7.  \((47a^{18}b^{23}c^5)^0\)

    Answer

    \(1\)

    8.  \(4^{−1}\)

    9.  \((2y)^{−3}\)

    Answer

    \(\dfrac{1}{8y^3}\)

    10.  \(p^{−3}·p^{−8}\)

    11.  \(\dfrac{x^4}{x^{−5}}\)

    Answer

    \(x^9\)

    12.  \((3x^{−3})^2\)

    13.  \(\dfrac{24r^3s}{6r^2s^7}\)

    Answer

    \(\dfrac{4r}{s^6}\)

    14.  \((x4y9x−3)2\)

    15.  \((8xy^3)(−6x^4y^6)\)

    Answer

    \(−48x^5y^9\)

    16.  \(4u(u^2−9u+1)\)

    17.  \((m+3)(7m−2)\)

    Answer

    \(21m^2−19m−6\)

    18.  \((n−8)(n^2−4n+11)\)

    19.  \((4x−3)^2\)

    Answer

    \(16x^2−24x+9\)

    20.  \((5x+2y)(5x−2y)\)

    21.  \((15xy^3−35x^2y)÷5xy\)

    Answer

    \(3y^2−7x \)

    22.  \((3x^3−10x^2+7x+10)÷(3x+2)\)

    23. Use the Factor Theorem to determine if \(x+3\) a factor of \(x^3+8x^2+21x+18\).

    Answer

    yes

    24.  a. Convert 112,000 to scientific notation.
          b. Convert \(5.25×10^{−4}\) to decimal form.

    In the following exercises, simplify and write your answer in decimal form.

    25.  \((2.4×10^8)(2×10^{−5})\)

    Answer

    \(4.4×10^3\)

    26.  \(\dfrac{9×10^4}{3×10^{−1}}\)

    27. For the function \(f(x)=6x^2−3x−9\) find:
         a. \(f(3)\)    b. \(f(−2)\)    c. \(f(0)\)

    Answer

    a. \(36\)    b. \(21\)    c. \(-9\)

    28. For \(f(x)=2x^2−3x−5\) and \(g(x)=3x^2−4x+1\), find
         a. \((f+g)(x)\)    b. \((f+g)(1)\)
         c. \((f−g)(x)\)    d. \((f−g)(−2)\)

    29. For functions \(f(x)=3x^2−23x−36\) and \(g(x)=x−9\), find
         a. \(\left(\dfrac{f}{g}\right)(x)\)    b. \(\left(\dfrac{f}{g}\right)(3)\)

    Answer

    a. \(\left(\dfrac{f}{g}\right)(x)=3x+4\)
    b. \(\left(\dfrac{f}{g}\right)(3)=13\)

    30. A hiker drops a pebble from a bridge \(240\) feet above a canyon. The function \(h(t)=−16t^2+240\) gives the height of the pebble \(t\) seconds after it was dropped. Find the height when \(t=3\).