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10.3: Exercises

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    Exercise \(\PageIndex{1}\)

    1. Find all rational roots of \(f(x)=2x^3-3x^2-3x+2\).
    2. Find all rational roots of \(f(x)=3x^3-x^2+15x-5\).
    3. Find all rational roots of \(f(x)=6x^3+7x^2-11x-12\).
    4. Find all real roots of \(f(x)=6x^4+25x^3+8x^2-7x-2\).
    5. Find all real roots of \(f(x)=4x^3+9x^2+26x+6\).
    Answer
    1. \(x=-1, x=2, x=\dfrac{1}{2}\)
    2. \(x=\dfrac{1}{3}\)
    3. \(x=\dfrac{-3}{2}, x=-1, x=\dfrac{4}{3},\)
    4. \(x=\dfrac{1}{2}, x=\dfrac{-2}{3}, x=-2+\sqrt{3}, x=-2-\sqrt{3}\)
    5. \(x=-\dfrac{1}{4}\)

    Exercise \(\PageIndex{2}\)

    Find a root of the polynomial by guessing possible candidates of the root.

    1. \(f(x)=x^5-1\)
    2. \(f(x)=x^4-1\)
    3. \(f(x)=x^3-27\)
    4. \(f(x)=x^3+1000\)
    5. \(f(x)=x^4-81\)
    6. \(f(x)=x^3-125\)
    7. \(f(x)=x^{5}+32\)
    8. \(f(x)=x^{777}-1\)
    9. \(f(x)=x^2+64\)
    Answer
    1. \(x = 1\)
    2. \(x = 1\) or \(x = −1\)
    3. \(x = 3\)
    4. \(x = −10\)
    5. \(x = 3\) or \(x = −3\)
    6. \(x = 5\)
    7. \(x = −2\)
    8. \(x = 1\)
    9. \(x = 8i\) or \(x = −8i\)

    Exercise \(\PageIndex{3}\)

    Find the roots of the polynomial and use it to factor the polynomial completely.

    1. \(f(x)=x^3-7x+6\)
    2. \(f(x)=x^3-x^2-16x-20\)
    3. \(f(x)=x^4-5x^2+4\)
    4. \(f(x)=x^3+x^2-5x-2\)
    5. \(f(x)=2x^3+x^2-7x-6\)
    6. \(f(x)=12x^3+49x^2-2x-24\)
    7. \(f(x)=x^4-1\)
    8. \(f(x)=x^5-6x^4+8x^3+6x^2-9x\)
    9. \(f(x)=x^3-27\)
    10. \(f(x)=x^4+2x^2-15\)
    Answer
    1. \(f(x)=(x-2)(x-1)(x+3)\)
    2. \(f(x)=(x-5)(x+2)^{2}\)
    3. \(f(x)=(x-1)(x+1)(x-2)(x+2)\)
    4. \(f(x)=(x-2)\left(x-\dfrac{-3+\sqrt{5}}{2}\right)\left(x-\dfrac{-3-\sqrt{5}}{2}\right)\)
    5. \(f(x)=2\left(x+\dfrac{3}{2}\right)(x+1)(x-2)\)
    6. \(f(x)=12\left(x-\dfrac{2}{3}\right)\left(x+\dfrac{3}{4}\right)(x+4)\)
    7. \(f(x)=(x-1)(x+1)(x-i)(x+i)\)
    8. \(f(x)=x(x-1)(x+1)(x-3)^{2}\)
    9. \(f(x)=(x-3)\left(x-\dfrac{-3+3 \sqrt{3} \cdot i}{2}\right)\left(x-\dfrac{-3-3 \sqrt{3} \cdot i}{2}\right)\)
    10. \(f(x)=(x-\sqrt{3})(x+\sqrt{3})(x-\sqrt{5} \cdot i)(x+\sqrt{5} \cdot i)\)

    Exercise \(\PageIndex{4}\)

    Find the exact roots of the polynomial; write the roots in simplest radical form, if necessary. Sketch a graph of the polynomial with all roots clearly marked.

    1. \(f(x)=x^3-2x^2-5x+6\)
    2. \(f(x)=x^3+5x^2+3x-4\)
    3. \(f(x)=-x^3+5x^2+7x-35\)
    4. \(f(x)=x^3+7x^2+13x+7\)
    5. \(f(x)=2x^3-8x^2-18x-36\)
    6. \(f(x)=x^4-4x^2+3\)
    7. \(f(x)=-x^4+x^3+24x^2-4x-80\)
    8. \(f(x)=7x^3-11x^2-10x+8\)
    9. \(f(x)=-15x^3+41x^2+15x-9\)
    10. \(f(x)=x^4-6x^3+6x^2+4x\)
    Answer
    1. clipboard_e9e588e98b98c48bdd3c8195ca7fda394.png
    2. clipboard_e574aa6bb305523aeb7a8835375039c1b.png
    3. clipboard_e47ad297c187d82fbadad1c9688721edf.png
    4. clipboard_e59028f639f84e64c9024e4e4142b2e36.png
    5. clipboard_eb5414325803b7b2e37c982f979c093f5.png
    6. clipboard_ec5eb742ef834999cda4b878539fb60e7.png
    7. clipboard_ec084b5d66298a5770cc7b41c351558fe.png
    8. clipboard_e7da7ee7eb592a36fee8cfd42ab317601.png
    9. clipboard_e852419d382a7b16d294c52a0e6f3156d.png
    10. clipboard_e3b847d557ffa6b84a9439be562ff9f5d.png

    Exercise \(\PageIndex{5}\)

    Find a polynomial \(f\) that fits the given data.

    1. \(f\) has degree \(3\). The roots of \(f\) are precisely \(2\), \(3\), \(4\). The leading coefficient of \(f\) is \(2\).
    2. \(f\) has degree \(4\). The roots of \(f\) are precisely \(-1\), \(2\), \(0\), \(-3\). The leading coefficient of \(f\) is \(-1\).
    3. \(f\) has degree \(3\). \(f\) has roots \(-2\), \(-1\), \(2\), and \(f(0)=10\).
    4. \(f\) has degree \(4\). \(f\) has roots \(0\), \(2\), \(-1\), \(-4\), and \(f(1)=20\).
    5. \(f\) has degree \(3\). The coefficients of \(f\) are all real. The roots of \(f\) are precisely \(2+5i\), \(2-5i\), \(7\). The leading coefficient of \(f\) is \(3\).
    6. \(f\) has degree \(3\). The coefficients of \(f\) are all real. \(f\) has roots \(i\), \(3\), and \(f(0)=6\).
    7. \(f\) has degree \(4\). The coefficients of \(f\) are all real. \(f\) has roots \(5+i\) and \(5-i\) of multiplicity \(1\), the root \(3\) of multiplicity \(2\), and \(f(5)=7\).
    8. \(f\) has degree \(4\). The coefficients of \(f\) are all real. \(f\) has roots \(i\) and \(3+2i\).
    9. \(f\) has degree \(6\). \(f\) has complex coefficients. \(f\) has roots \(1+i\), \(2+i\), \(4-3i\) of multiplicity \(1\) and the root \(-2\) of multiplicity \(3\).
    10. \(f\) has degree \(5\). \(f\) has complex coefficients. \(f\) has roots \(i\), \(3\), \(-7\) (and possibly other roots).
    11. \(f\) has degree \(3\). The roots of \(f\) are determined by its graph:

    clipboard_ed02f942d9bdafc3eedf9e98f36a5ac5d.png

    1. \(f\) has degree \(4\). The coefficients of \(f\) are all real. The leading coefficient of \(f\) is 1. The roots of \(f\) are determined by its graph: (see Section 9.3).

    clipboard_e627fb6f60fa1dd609266bab42e4f6a6c.png

    1. \(f\) has degree \(4\). The coefficients of \(f\) are all real. \(f\) has the following graph:

    clipboard_e845de0dad18ac50917a919fc0918c97a.png

    Answer
    1. \(f(x)=2(x-2)(x-3)(x-4)\)
    2. \(f(x)=(-1) \cdot x(x-2)(x+1)(x+3)\)
    3. \(f(x)=\left(-\dfrac{5}{2}\right) \cdot(x-2)(x+2)(x+1)\)
    4. \(f(x)=-2 \cdot x(x-2)(x+1)(x+4)\)
    5. \(f(x)=3(x-7)(x-(2+5 i))(x-(2-5 i))\)
    6. \(f(x)=(-2) \cdot(x-i)(x+i)(x-3)\)
    7. \(f(x)=\frac{7}{4} \cdot(x-(5+i))(x-(5-i))(x-3)^{2}\)
    8. \(f(x)=(x-i)(x+i)(x-(3+2 i))(x-(3-2 i))\) (other correct answers are possible, depending on the choice of the first coefficient)
    9. \(f(x)=(x-(1+i))(x-(2+i))(x-(4-3 i))(x+2)^{3}\) (other correct answers are possible, depending on the choice of the first coefficient)
    10. \(f(x)=(x-i)(x-3)(x+7)^{2}\) (other correct answers are possible, depending on the choice of the first coefficient and the fourth root)
    11. \(f(x)=(x-2)(x-3)(x-4)\) (other correct answers are possible, depending on the choice of the first coefficient)
    12. \(f(x)=(x-1)^{2}(x-3)^{2}\)
    13. \(f(x)=-x(x-1)(x-3)(x-4)\) (other correct answers are possible, depending on the choice of the first coefficient)

    This page titled 10.3: 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; a detailed edit history is available upon request.