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5.E: Review Exercises and Sample Exam

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    Review Exercises

    Exercise \(\PageIndex{1}\) Rules of Exponents

    Simplify.

    1. \(7^{3}⋅7^{6}\)
    2. \(5^{9}5^{6}\)
    3. \(y^{5}⋅y^{2}⋅y^{3}\)
    4. \(x^{3}y^{2}⋅xy^{3}\)
    5. \(−5a^{3}b^{2}c⋅6a^{2}bc^{2}\)
    6. \(\frac{55x^{2}yz}{55xyz^{2}}\)
    7. \((\frac{−3a^{2}b^{4}}{2c^{3}})^{2}\)
    8. \((−2a^{3}b^{4}c^{4})^{3}\)
    9. \(−5x^{3}y^{0}(z^{2})^{3}⋅2x^{4}(y^{3})^{2}z\)
    10. \((−25x^{6}y^{5}z)^{0}\)
    11. Each side of a square measures \(5x^{2}\) units. Find the area of the square in terms of \(x\).
    12. Each side of a cube measures \(2x^{3}\) units. Find the volume of the cube in terms of \(x\).
    Answer

    1. \(7^{9}\)

    3. \(y^{10}\)

    5. \(−30a^{5}b^{3}c^{3}\)

    7. \(\frac{9a^{4}b^{8}}{4c^{6}}\)

    9. \(−10x^{7}y^{6}z^{7}\)

    11. \(A=25x^{4}\)

    Exercise \(\PageIndex{2}\) Introduction to Polynomials

    Classify the given polynomial as a monomial, binomial, or trinomial and state the degree.

    1. \(8a^{3}−1\)
    2. \(5y^{2}−y+1\)
    3. \(−12ab^{2}\)
    4. \(10\)
    Answer

    1. Binomial; degree \(3\)

    3. Monomial; degree \(3\)

    Exercise \(\PageIndex{3}\) Introduction to Polynomials

    Write the following polynomials in standard form.

    1. \(7−x^{2}−5x\)
    2. \(5x^{2}−1−3x+2x^{3}\)
    Answer

    1. \(-x^{2}-5x+7\)

    Exercise \(\PageIndex{4}\) Introduction to Polynomials

    Evaluate.

    1. \(2x^{2}−x+1\), where \(x=−3\)
    2. \(\frac{1}{2}x−\frac{3}{4}\), where \(x=\frac{1}{3}\)
    3. \(b^{2}−4ac\), where \(a=−\frac{1}{2}, b=−3\), and \(c=−\frac{3}{2}\)
    4. \(a^{2}−b^{2}\), where \(a=−\frac{1}{2}\) and \(b=−\frac{1}{3}\)
    5. \(a^{3}−b^{3}\), where \(a=−2\) and \(b=−1\)
    6. \(xy^{2}−2x^{2}y\), where \(x=−3\) and \(y=−1\)
    7. Given \(f(x)=3x^{2}−5x+2\), find \(f(−2)\).
    8. Given \(g(x)=x^{3}−x^{2}+x−1\), find \(g(−1)\).
    9. The surface area of a rectangular solid is given by the formula \(SA=2lw+2wh+2lh\), where \(l, w\), and \(h\) represent the length, width, and height, respectively. If the length of a rectangular solid measures \(2\) units, the width measures \(3\) units, and the height measures \(5\) units, then calculate the surface area.
    10. The surface area of a sphere is given by the formula \(SA=4πr^{2}\), where \(r\) represents the radius of the sphere. If a sphere has a radius of \(5\) units, then calculate the surface area.
    Answer

    1. \(22\)

    3. \(6\)

    5. \(−7\)

    7. \(f(−2)=24\)

    9. \(62\) square units

    Exercise \(\PageIndex{5}\) Adding and Subtracting Polynomials

    Perform the operations.

    1. \((3x−4)+(9x−1)\)
    2. \((13x−19)+(16x+12)\)
    3. \((7x^{2}−x+9)+(x^{2}−5x+6)\)
    4. \((6x^{2}y−5xy^{2}−3)+(−2x^{2}y+3xy^{2}+1)\)
    5. \((4y+7)−(6y−2)+(10y−1)\)
    6. \((5y^{2}−3y+1)−(8y^{2}+6y−11)\)
    7. \((7x^{2}y^{2}−3xy+6)−(6x^{2}y^{2}+2xy−1)\)
    8. \((a^{3}−b^{3})−(a^{3}+1)−(b^{3}−1)\)
    9. \((x^{5}−x^{3}+x−1)−(x^{4}−x^{2}+5)\)
    10. \((5x^{3}−4x^{2}+x−3)−(5x^{3}−3)+(4x^{2}−x)\)
    11. Subtract \(2x−1\) from \(9x+8\).
    12. Subtract \(3x^{2}−10x−2\) from \(5x^{2}+x−5\).
    13. Given \(f(x)=3x^{2}−x+5\) and \(g(x)=x^{2}−9\), find \((f+g)(x)\).
    14. Given \(f(x)=3x^{2}−x+5\) and \(g(x)=x^{2}−9\), find \((f−g)(x)\).
    15. Given \(f(x)=3x^{2}−x+5\) and \(g(x)=x^{2}−9\), find \((f+g)(−2)\).
    16. Given \(f(x)=3x^{2}−x+5\) and \(g(x)=x^{2}−9\), find \((f−g)(−2)\).
    Answer

    1. \(12x−5\)

    3. \(8x^{2}−6x+15\)

    5. \(8y+8\)

    7. \(x^{2}y^{2}−5xy+7\)

    9. \(x^{5}−x^{4}−x^{3}+x^{2}+x−6\)

    11. \(7x+9\)

    13. \((f+g)(x)=4x^{2}−x−4\)

    15. \((f+g)(−2)=14\)

    Exercise \(\PageIndex{6}\) Multiplying Polynomials

    Multiply.

    1. \(6x^{2}(−5x^{4})\)
    2. \(3ab^{2}(7a^{2}b)\)
    3. \(2y(5y−12)\)
    4. \(−3x(3x^{2}−x+2)\)
    5. \(x^{2}y(2x^{2}y−5xy^{2}+2)\)
    6. \(−4ab(a^{2}−8ab+b^{2})\)
    7. \((x−8)(x+5)\)
    8. \((2y−5)(2y+5)\)
    9. \((3x−1)^{2}\)
    10. \((3x−1)^{3}\)
    11. \((2x−1)(5x^{2}−3x+1)\)
    12. \((x^{2}+3)(x^{3}−2x−1)\)
    13. \((5y+7)^{2}\)
    14. \((y^{2}−1)^{2}\)
    15. Find the product of \(x^{2}−1\) and \(x^{2}+1\).
    16. Find the product of \(32x^{2}y\) and \(10x−30y+2\).
    17. Given \(f(x)=7x−2\) and \(g(x)=x^{2}−3x+1\), find \((f⋅g)(x)\).
    18. Given \(f(x)=x−5\) and \(g(x)=x^{2}−9\), find \((f⋅g)(x)\).
    19. Given \(f(x)=7x−2\) and \(g(x)=x^{2}−3x+1\), find \((f⋅g)(−1)\).
    20. Given \(f(x)=x−5\) and \(g(x)=x^{2}−9\), find \((f⋅g)(−1)\).
    Answer

    1. \(−30x^{6}\)

    3. \(10y^{2}−24y\)

    5. \(2x^{4}y^{2}−5x^{3}y^{3}+2x^{2}y\)

    7. \(x^{2}−3x−40\)

    9. \(9x^{2}−6x+1\)

    11. \(10x^{3}−11x^{2}+5x−1\)

    13. \(25y^{2}+70y+49\)

    15. \(x^{4}−1\)

    17. \((f⋅g)(x)=7x^{3}−23x^{2}+13x−2\)

    19. \((f⋅g)(−1)=−45\)

    Exercise \(\PageIndex{7}\) Dividing Polynomials

    Divide.

    1. \(\frac{7y^{2}−14y+28}{7}\)
    2. \(\frac{12x^{5}−30x^{3}+6x}{6x}\)
    3. \(\frac{4a^{2}b−16ab^{2}−4ab}{−4ab}\)
    4. \(\frac{6a^{6}−24a^{4}+5a^{2}}{3a^{2}}\)
    5. \((10x^{2}−19x+6)÷(2x−3)\)
    6. \((2x^{3}−5x^{2}+5x−6)÷(x−2) \)
    7. \(\frac{10x^{4}−21x^{3}−16x^{2}+23x−20}{2x−5}\)
    8. \(\frac{x^{5}−3x^{4}−28x^{3}+61x^{2}−12x+36}{x−6}\)
    9. \(\frac{10x^{3}−55x^{2}+72x−4}{2x−7}\)
    10. \(\frac{3x^{4}+19x^{3}+3x^{2}−16x−11}{3x+1}\)
    11. \(\frac{5x^{4}+4x^{3}−5x^{2}+21x+21}{5x+4}\)
    12. \(\frac{x^{4}−4}{x−4}\)
    13. \(\frac{2x^{4}+10x^{3}−23x^{2}−15x+30}{2x^{2}−3}\)
    14. \(\frac{7x^{4}−17x^{3}+17x^{2}−11x+2}{x^{2}−2x+1}\)
    15. Given \(f(x)=x^{3}−4x+1\) and \(g(x)=x−1\), find \((f/g)(x)\).
    16. Given \(f(x)=x^{5}−32\) and \(g(x)=x−2\), find \((f/g)(x)\).
    17. Given \(f(x)=x^{3}−4x+1\) and \(g(x)=x−1\), find \((f/g)(2)\).
    18. Given \(f(x)=x^{5}−32\) and \(g(x)=x−2\), find \((f/g)(0)\).
    Answer

    1. \(y^{2}−2y+4\)

    3. \(−a+4b+1\)

    5. \(5x−2\)

    7. \(5x^{3}+2x^{2}−3x+4\)

    9. \(5x^{2}−10x+1+\frac{3}{2x−7}\)

    11. \(x^{3}−x+5+\frac{1}{5x+4}\)

    13. \(x^{2}+5x−10\)

    15. \((f/g)(x)=x^{2}+x−3−\frac{2}{x−1}\)

    17. \((f/g)(2)=1\)

    Exercise \(\PageIndex{8}\) Negative Exponents

    Simplify.

    1. \((−10)^{−2}\)
    2. \(−10^{−2}\)
    3. \(5x^{−3}\)
    4. \((5x)^{−3}\)
    5. \(\frac{1}{7y^{-3}}\)
    6. \(3x^{−4}y^{−2}\)
    7. \(\frac{−2a^{2}b^{−5}}{c^{−8}}\)
    8. \((−5x^{2}yz^{−1})^{−2}\)
    9. \((−2x^{−3}y^{0}z^{2})^{−3}\)
    10. \((\frac{−10a^{5}b^{3}c^{2}}{5ab^{2}c^{2}})^{−1}\)
    11. \((\frac{a^{2}b^{−4}c^{0}}{2a^{4}b^{−3}c})^{−3}\)
    Answer

    1. \(\frac{1}{100}\)

    3. \(\frac{5}{x^{3}}\)

    5. \(\frac{y^{3}}{7}\)

    7. \(\frac{−2a^{2}c^{8}}{b^{5}}\)

    9. \(\frac{−x^{9}}{8z^{6}}\)

    11. \(8a^{6}b^{3}c^{3}\)

    Exercise \(\PageIndex{9}\) Negative Exponents

    The value in dollars of a new laptop computer can be estimated by using the formula \(V=1200(t+1)^{−1}\), where \(t\) represents the number of years after the purchase.

    1. Estimate the value of the laptop when it is \(1\frac{1}{2}\) years old.
    2. What was the laptop worth new?
    Answer

    2. $\(1,200\)

    Exercise \(\PageIndex{10}\) Negative Exponents

    Rewrite using scientific notation.

    1. \(2,030,000,000\)
    2. \(0.00000004011\)
    Answer

    2. \(5.796×10^{19}\)

    Exercise \(\PageIndex{11}\) Negative Exponents

    Perform the indicated operations.

    1. \((5.2×10^{12})(1.8×10^{−3})\)
    2. \((9.2×10^{−4})(6.3×10^{22})\)
    3. \(\frac{4×10^{16}}{8×10^{−7}}\)
    4. \(\frac{9×10^{−30}}{4×10^{−10}}\)
    5. \(5,000,000,000,000 × 0.0000023\)
    6. \(\frac{0.0003}{120,000,000,000,000}\)
    Answer

    2. \(5.796×10^{19}\)

    4. \(2.25×10^{−20}\)

    6. \(2.5×10^{−18}\)

    Simple Exam

    Exercise \(\PageIndex{12}\)

    Simplify.

    1. \(−5x^{3}(2x^{2}y)\)
    2. \((x^{2})^{4}⋅x^{3}⋅x\)
    3. \(\frac{(−2x^{2}y^{3})^{2}}{x^{2}y}\)
      1. \((−5)^{0}\)
      2. \(−5^{0}\)
    Answer

    1. \(−10x^{5}y\)

    3. \(4x^{2}y^{5}\)

    Exercise \(\PageIndex{13}\)

    Evaluate.

    1. \(2x^{2}−x+5\), where \(x=−5\)
    2. \(a^{2}−b^{2}\), where \(a=4\) and \(b=−3\)
    Answer

    1. \(60\)

    Exercise \(\PageIndex{14}\)

    Perform the operations.

    1. \((3x^{2}−4x+5)+(−7x^{2}+9x−2) \)
    2. \((8x^{2}−5x+1)−(10x^{2}+2x−1) \)
    3. \((\frac{3}{5}a−\frac{1}{2})−(\frac{2}{3}a^{2}+\frac{2}{3}a−\frac{2}{9})+(\frac{1}{15}a−\frac{5}{18})\)
    4. \(2x^{2}(2x^{3}−3x^{2}−4x+5)\)
    5. \((2x−3)(x+5)\)
    6. \((x−1)^{3}\)
    7. \(\frac{81x^{5}y^{2}z}{-3x^{3}yz}\)
    8. \(\frac{10x^{9}−15x^{5}+5x^{2}}{−5x^{2}}\)
    9. \(\frac{x^{3}−5x^{2}+7x−2}{x−2}\)
    10. \(\frac{6x^{4}−x^{3}−13x^{2}−2x−1}{2x−1}\)
    Answer

    1. \(−4x^{2}+5x+3 \)

    3. \(−\frac{2}{3}a^{2}−\frac{5}{9}\)

    5. \(2x^{2}+7x−15 \)

    7. \(−27x^{2}y\)

    9. \(x^{2}−3x+1\)

    Exercise \(\PageIndex{15}\)

    Simplify.

    1. \(2^{−3}\)
    2. \(−5x^{−2}\)
    3. \((2x^{4}y^{−3}z)^{−2}\)
    4. \((\frac{−2a^{3}b^{−5}c^{−2}}{ab^{−3}c^{2}})^{−3}\)
    5. Subtract \(5x^{2}y−4xy^{2}+1\) from \(10x^{2}y−6xy^{2}+2\).
    6. If each side of a cube measures \(4x4\) units, calculate the volume in terms of \(x\).
    7. The height of a projectile in feet is given by the formula \(h=−16t^{2}+96t+10\), where \(t\) represents time in seconds. Calculate the height of the projectile at \(1\frac{1}{2}\) seconds.
    8. The cost in dollars of producing custom t-shirts is given by the formula \(C=120+3.50x\), where \(x\) represents the number of t-shirts produced. The revenue generated by selling the t-shirts for $\(6.50\) each is given by the formula \(R=6.50x\), where \(x\) represents the number of t-shirts sold.
      1. Find a formula for the profit. (profit = revenue − cost)
      2. Use the formula to calculate the profit from producing and selling \(150\) t-shirts.
    9. The total volume of water in earth’s oceans, seas, and bays is estimated to be \(4.73×10^{19}\) cubic feet. By what factor is the volume of the moon, \(7.76×10^{20}\) cubic feet, larger than the volume of earth’s oceans? Round to the nearest tenth.
    Answer

    1. \(\frac{1}{8}\)

    3. \(\frac{y^{6}}{4x^{8}z^{2}}\)

    5. \(5x^{2}y−2xy^{2}+1\)

    7. \(118\) feet

    9. \(16.4\)

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