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8.4: Exercises

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    Exercise \(\PageIndex{1}\)

    Divide by long division.

    1. \(\dfrac{x^3-4x^2+2x+1}{x-2}\)
    2. \(\dfrac{x^3+6x^2+7x-2}{x+3}\)
    3. \(\dfrac{x^2+7x-4}{x+1}\)
    4. \(\dfrac{x^3+3x^2+2x+5}{x+2}\)
    5. \(\dfrac{2x^3+x^2+3x+5}{x-1}\)
    6. \(\dfrac{2x^4+7x^3+x+3}{x+5}\)
    7. \(\dfrac{2x^4-31x^2-13}{x-4}\)
    8. \(\dfrac{x^3+27}{x+3}\)
    9. \(\dfrac{3x^4+7x^3+5x^2+7x+4}{3x+1}\)
    10. \(\dfrac{8x^3+18x^2+21x+18}{2x+3}\)
    11. \(\dfrac{x^3+3x^2-4x-5}{x^2+2x+1}\)
    12. \(\dfrac{x^5+3x^4-20}{x^2+3}\)
    Answer
    1. \(x^{2}-2 x-2-\dfrac{3}{x-2}\)
    2. \(x^{2}+3 x-2+\dfrac{4}{x+3}\)
    3. \(x+6-\dfrac{10}{x+1}\)
    4. \(x^{2}+x+\dfrac{5}{x+2}\)
    5. \(2 x^{2}+3 x+6+\dfrac{11}{x-1}\)
    6. \(2 x^{3}-3 x^{2}+15 x-74+\dfrac{373}{x+5}\)
    7. \(2 x^{3}+8 x^{2}+x+4+\dfrac{3}{x-4}\)
    8. \(x^{2}-3 x+9\)
    9. \(x^{3}+2 x^{2}+x+2+\dfrac{2}{3 x+1}\)
    10. \(4 x^{2}+3 x+6\)
    11. \(x+1-\dfrac{7 x+6}{x^{2}+2 x+1}\)
    12. \(x^{3}+3 x^{2}-3 x-9+\dfrac{9 x+7}{x^{2}+3}\)

    Exercise \(\PageIndex{2}\)

    Find the remainder when dividing \(f(x)\) by \(g(x)\).

    1. \(f(x)=x^3+2x^2+x-3, \quad g(x)=x-2\)
    2. \(f(x)=x^3-5x+8, \quad g(x)=x-3\)
    3. \(f(x)=x^5-1, \quad g(x)=x+1\)
    4. \(f(x)=x^5+5x^2-7x+10, \quad g(x)=x+2\)
    Answer
    1. remainder \(r = 15\)
    2. \(r = 20\)
    3. \(r = -2\)
    4. \(r = 12\)

    Exercise \(\PageIndex{3}\)

    Determine whether the given \(g(x)\) is a factor of \(f(x)\). If so, name the corresponding root of \(f(x)\).

    1. \(f(x)=x^2+5x+6, \quad g(x)=x+3\)
    2. \(f(x)=x^3-x^2-3x+8, \quad g(x)=x-4\)
    3. \(f(x)=x^4+7x^3+3x^2+29x+56, \quad g(x)=x+7\)
    4. \(f(x)=x^{999}+1, \quad g(x)=x+1\)
    Answer
    1. yes, \(g(x)\) is a factor of \(f(x)\), the root of \(f(x)\) is \(x = −3\)
    2. \(g(x)\) is not a factor of \(f(x)\)
    3. \(g(x)\) is a factor of \(f(x)\), the root of \(f(x)\) is \(x = −7\)
    4. \(g(x)\) is a factor of \(f(x)\), the root of \(f(x)\) is \(x = −1\)

    Exercise \(\PageIndex{4}\)

    Check that the given numbers for \(x\) are roots of \(f(x)\) (see Observation). If the numbers \(x\) are indeed roots, then use this information to factor \(f(x)\) as much as possible.

    1. \(f(x)=x^3-2x^2-x+2, \quad x=1\)
    2. \(f(x)=x^3-6x^2+11x-6, \quad x=1, x=2, x=3\)
    3. \(f(x)=x^3-3x^2+x-3, \quad x=3\)
    4. \(f(x)=x^3+6x^2+12x+8, \quad x=-2\)
    5. \(f(x)=x^3+13x^2+50x+56, \quad x=-3, x=-4\)
    6. \(f(x)=x^3+3x^2-16x-48, \quad x=2, x=-4\)
    7. \(f(x)=x^5+5x^4-5x^3-25x^2+4x+20, \quad x=1, x=-1, \quad x=2, x=-2\)
    Answer
    1. \(f(x)=(x-2)(x-1)(x+1)\)
    2. \(f(x)=(x-1)(x-2)(x-3)\)
    3. \(f(x)=(x-3)(x-i)(x+i)\)
    4. \(f(x)=(x+2)^{3}\)
    5. \(f(x)=(x+2)(x+4)(x+7)\)
    6. \(f(x)=(x-4)(x+3)(x+4)\)
    7. \(f(x)=(x-2)(x-1)(x+1)(x+2)(x+5)\)

    Exercise \(\PageIndex{5}\)

    Divide by using synthetic division.

    1. \(\dfrac{2x^3+3x^2-5x+7}{x-2}\)
    2. \(\dfrac{4x^3+3x^2-15x+18}{x+3}\)
    3. \(\dfrac{x^3+4x^2-3x+1}{x+2}\)
    4. \(\dfrac{x^4+x^3+1}{x-1}\)
    5. \(\dfrac{x^5+32}{x+2}\)
    6. \(\dfrac{x^3+5x^2-3x-10}{x+5}\)
    Answer
    1. \(2 x^{2}+7 x+9+\dfrac{25}{x-2}\)
    2. \(4 x^{2}-9 x+12-\dfrac{18}{x+3}\)
    3. \(x^{2}+2 x-7+\dfrac{15}{x+2}\)
    4. \(x^{3}+2 x^{2}+2 x+2+\dfrac{3}{x-1}\)
    5. \(x^{4}-2 x^{3}+4 x^{2}-8 x+16\)
    6. \(x^{2}-3+\dfrac{5}{x+5}\)

    This page titled 8.4: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Thomas Tradler and Holly Carley (New York City College of Technology at CUNY Academic Works) via source content that was edited to the style and standards of the LibreTexts platform.