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27.2: Review of polynomials and rational functions

  • Page ID
    68483
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    Exercise \(\PageIndex{1}\)

    Divide the polynomials: \(\dfrac{2x^3+x^2-9x-8}{2x+3}\)

    Answer

    \(x^{2}-x-3+\dfrac{1}{2 x+3}\)

    Exercise \(\PageIndex{2}\)

    Find the remainder when dividing \(x^3+3x^2-5x+7\) by \(x+2\).

    Answer

    \(21\)

    Exercise \(\PageIndex{3}\)

    Which of the following is a factor of \(x^{400}-2x^{99}+1\): \[x-1, \quad x+1, \quad x-0 \nonumber \]

    Answer

    \(x − 1\) is a factor, \(x + 1\) is not a factor, \(x − 0\) is not a factor

    Exercise \(\PageIndex{4}\)

    Identify the polynomial with its graph.

    1. clipboard_e0c0295236f728781dd9f837be2a680f2.png
    2. clipboard_ea73579cf5c91f70adf630ab2ee0b3612.png
    3. clipboard_e86c1d73b604adf1df9c67be8ac6bb649.png
    4. clipboard_e1b082903b9681e6702d6ec155312e9db.png
    1. \(f(x)=-x^{2}+2 x+1\) graph: _______________
    2. \(f(x)=-x^{3}+3 x^{2}-3 x+2\) graph: _______________
    3. \(f(x)=x^{3}-3 x^{2}+3 x+1\) graph: _______________
    4. \(f(x)=x^{4}-4 x^{3}+6 x^{2}-4 x+2\) graph: _______________
    Answer
    1. \(\leftrightarrow \text { iii) }\)
    2. \(\leftrightarrow \text { iv) }\)
    3. \(\leftrightarrow \mathrm{i})\)
    4. \(\leftrightarrow \text { ii) }\)

    Exercise \(\PageIndex{5}\)

    Sketch the graph of the function: \[f(x)=x^4-10x^3-0.01x^2+0.1x \nonumber \]

    • What is your viewing window?
    • Find all roots, all maxima and all minima of the graph with the calculator.
    Answer

    clipboard_e204f31b4ec80b0feba7e396e7b6b5199.png

    Exercise \(\PageIndex{6}\)

    Find all roots of \(f(x)=x^3+6x^2+5x-12\).

    Use this information to factor \(f(x)\) completely.

    Answer

    \(f(x)=(x-1)(x+3)(x+4)\)

    Exercise \(\PageIndex{7}\)

    Find a polynomial of degree \(3\) whose roots are \(0\), \(1\), and \(3\), and so that \(f(2)=10\).

    Answer

    \(f(x)=(-5) \cdot x(x-1)(x-3)\)

    Exercise \(\PageIndex{8}\)

    Find a polynomial of degree \(4\) with real coefficients, whose roots include \(-2\), \(5\), and \(3-2i\).

    Answer

    \(f(x)=(x+2)(x-5)(x-(3-2 i))(x-(3+2 i))\) (other correct answers are possible, depending on the choice of the first coefficient)

    Exercise \(\PageIndex{9}\)

    Let \(f(x)=\dfrac{3x^2-12}{x^2-2x-3}\). Sketch the graph of \(f\). Include all vertical and horizontal asymptotes, all holes, and all \(x\)- and \(y\)-intercepts.

    Answer

    \(f(x)=\dfrac{3(x-2)(x+2)}{(x-3)(x+1)}\) has domain \(D=\mathbb{R}-\{-1,3\}\), horizontal asympt. \(y = 3\), vertical asympt. \(x = −1\) and \(x = 3\), no removable discont., \(x\)-intercepts at \(x = −2\) and \(x = 2\) and \(x = 3\), \(y\)-intercept at \(y = 4\), graph:

    clipboard_e28cf3f49f7fcd7d03fb0b4fd6823fa75.png

    Exercise \(\PageIndex{10}\)

    Solve for \(x\):

    1. \(x^4+2x< 2x^3+x^2\)
    2. \(x^2+3x\geq 7\)
    3. \(\dfrac{x+1}{x+4}\leq 2\)
    Answer
    1. \((-1,0) \cup(1,2)\)
    2. \(\left(-\infty, \dfrac{-3-\sqrt{37}}{2}\right] \cup\left[\dfrac{-3+\sqrt{37}}{2}, \infty\right)\)
    3. \((-\infty,-7] \cup(-4, \infty)\)

    This page titled 27.2: Review of polynomials and rational functions 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.