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Mathematics LibreTexts

3.R: Polynomial and Rational Functions(Review)

  • Page ID
    19674
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    3.1 Complex Numbers

    Perform the indicated operation with complex numbers.

    1) \((4+3 i)+(-2-5 i)\)

    Answer

    \(2-2 i\)

    2) \((6-5 i)-(10+3 i)\)

    3) \((2-3 i)(3+6 i)\)

    Answer

    \(24+3 i\)

    4) \(\dfrac{2-i}{2+i}\)

    Solve the following equations over the complex number system.

    5) \(x^{2}-4 x+5=0\)

    Answer

    \(\{2+i, 2-i\}\)

    6) \(x^{2}+2 x+10=0\)

    3.2 Quadratic Functions

    For the exercises 1-2, write the quadratic function in standard form. Then, give the vertex and axes intercepts. Finally, graph the function.

    1) \(f(x)=x^{2}-4 x-5\)

    Answer

    \(f(x)=(x-2)^{2}-9\) vertex \((2,-9)\), intercepts \((5,0); (-1,0); (0,-5)\)

    CNX_Precalc_Figure_03_09_206.jpg

    2) \(f(x)=-2 x^{2}-4 x\)

    For the problems 3-4, find the equation of the quadratic function using the given information.

    3) The vertex is \((-2,3)\) and a point on the graph is \((3,6)\).

    Answer

    \(f(x)=\dfrac{3}{25}(x+2)^{2}+3\)

    4) The vertex is \((-3,6.5)\) and a point on the graph is \((2,6)\).

    Answer the following questions.

    5) A rectangular plot of land is to be enclosed by fencing. One side is along a river and so needs no fence. If the total fencing available is \(600\) meters, find the dimensions of the plot to have maximum area.

    Answer

    \(300\) meters by \(150\) meters, the longer side parallel to river.

    6) An object projected from the ground at a \(45\) degree angle with initial velocity of \(120\) feet per second has height, \(h\), in terms of horizontal distance traveled, \(x\), given by \(h(x)=\dfrac{-32}{(120)^{2}} x^{2}+x\). Find the maximum height the object attains.

    3.3 Power Functions and Polynomial Functions

    For the exercises 1-3, determine if the function is a polynomial function and, if so, give the degree and leading coefficient.

    1) \(f(x)=4 x^{5}-3 x^{3}+2 x-1\)

    Answer

    Yes, \(\text{degree} = 5\), \(\text{leading coefficient} = 4\)

    2) \(f(x)=5^{x+1}-x^{2}\)

    3) \(f(x)=x^{2}\left(3-6 x+x^{2}\right)\)

    Answer

    Yes, \(\text{degree} = 4\), \(\text{leading coefficient} = 1\)

    For the exercises 4-6, determine end behavior of the polynomial function.

    4) \(f(x)=2 x^{4}+3 x^{3}-5 x^{2}+7\)

    5) \(f(x)=4 x^{3}-6 x^{2}+2\)

    Answer

    As \(x \rightarrow-\infty, f(x) \rightarrow-\infty \), as \(x \rightarrow \infty, f(x) \rightarrow \infty\)

    6) \(f(x)=2 x^{2}\left(1+3 x-x^{2}\right)\)

    3.4 Graphs of Polynomial Functions

    For the exercises 1-3, find all zeros of the polynomial function, noting multiplicities.

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

    Answer

    \(-3\) with multiplicity \(2\), \(-\dfrac{1}{2}\) with multiplicity \(1\), \(-1\) with multiplicity \(3\)

    2) \(f(x)=x^{5}+4 x^{4}+4 x^{3}\)

    3) \(f(x)=x^{3}-4 x^{2}+x-4\)

    Answer

    \(4\) with multiplicity \(1\)

    For the exercises 4-5, based on the given graph, determine the zeros of the function and note multiplicity.

    4)

    CNX_Precalc_Figure_03_09_208.jpg

    5)

    CNX_Precalc_Figure_03_09_209.jpg

    Answer

    \(\dfrac{1}{2}\) with multiplicity \(1\), \(3\) with multiplicity \(3\)

    6) Use the Intermediate Value Theorem to show that at least one zero lies between \(2\) and \(3\) for the function \(f(x)=x^{3}-5 x+1\)

    3.5 Dividing Polynomials

    For the exercises 1-2, use long division to find the quotient and remainder.

    1) \(\dfrac{x^{3}-2 x^{2}+4 x+4}{x-2}\)

    Answer

    \(x^{2}+4\) with remainder \(12\)

    2) \(\dfrac{3 x^{4}-4 x^{2}+4 x+8}{x+1}\)

    For the exercises 3-6, use synthetic division to find the quotient. If the divisor is a factor, then write the factored form.

    3) \(\dfrac{x^{2}-2 x^{2}+5 x-1}{x+3}\)

    Answer

    \(x^{2}-5 x+20-\dfrac{61}{x+3}\)

    4) \(\dfrac{x^{2}+4 x+10}{x-3}\)

    5) \(\dfrac{2 x^{3}+6 x^{2}-11 x-12}{x+4}\)

    Answer

    \(2 x^{2}-2x-3\), so factored form is \((x+4)\left(2 x^{2}-2x-3\right)\)

    6) \(\dfrac{3 x^{4}+3 x^{3}+2 x+2}{x+1}\)

    3.6 Zeros of Polynomial Functions

    For the exercises 1-4, use the Rational Zero Theorem to help you solve the polynomial equation.

    1) \(2 x^{3}-3 x^{2}-18 x-8=0\)

    Answer

    \(\left\{-2,4,-\dfrac{1}{2}\right\}\)

    2) \(3x^{3}+11 x^{2}+8 x-4=0\)

    3) \(2 x^{4}-17 x^{3}+46 x^{2}-43 x+12=0\)

    Answer

    \(\left\{1,3,4, \dfrac{1}{2}\right\}\)

    4) \(4 x^{4}+8 x^{3}+19 x^{2}+32 x+12=0\)

    For the exercises 5-6, use Descartes’ Rule of Signs to find the possible number of positive and negative solutions.

    5) \(x^{3}-3 x^{2}-2 x+4=0\)

    Answer

    \(0\) or \(2\) positive, \(1\) negative

    6) \(2 x^{4}-x^{3}+4 x^{2}-5 x+1=0\)

    3.7 Rational Functions

    For the following rational functions 1-4, find the intercepts and the vertical and horizontal asymptotes, and then use them to sketch a graph.

    1) \(f(x)=\dfrac{x+2}{x-5}\)

    Answer

    Intercepts \((-2,0)\) and \(\left(0,-\dfrac{2}{5}\right)\), Asymptotes \(x=5\) and \(y=1\)

    CNX_Precalc_Figure_03_09_210.jpg

    2) \(f(x)=\dfrac{x^{2}+1}{x^{2}-4}\)

    3) \(f(x)=\dfrac{3 x^{2}-27}{x^{2}-9}\)

    Answer

    Intercepts \((3,0),(-3,0)\), and \(\left(0, \dfrac{27}{2}\right)\), Asymptotes \(x=1, x=-2, y=3\)

    CNX_Precalc_Figure_03_09_212.jpg

    4) \(f(x)=\dfrac{x+2}{x^{2}-9}\)

    For the exercises 5-6, find the slant asymptote.

    5) \(f(x)=\dfrac{x^{2}-1}{x+2}\)

    Answer

    \(y=x-2\)

    6) \(f(x)=\dfrac{2 x^{3}-x^{2}+4}{x^{2}+1}\)

    3.8 Inverses and Radical Functions

    For the exercises 1-6, find the inverse of the function with the domain given.

    1) \(f(x)=(x-2)^{2}, x \geq 2\)

    Answer

    \(f^{-1}(x)=\sqrt{x}+2\)

    2) \(f(x)=(x+4)^{2}-3, x \geq-4\)

    3) \(f(x)=x^{2}+6 x-2, x \geq-3\)

    Answer

    \(f^{-1}(x)=\sqrt{x+11}-3\)

    4) \(f(x)=2 x^{3}-3\)

    5) \(f(x)=\sqrt{4 x+5}-3\)

    Answer

    \(f^{-1}(x)=\dfrac{(x+3)^{2}-5}{4}, x \geq-3\)

    6) \(f(x)=\dfrac{x-3}{2 x+1}\)

    3.9 Modeling Using Variation

    For the exercises 1-4, find the unknown value.

    1) \(y\) varies directly as the square of \(x\). If when \(x=3, y=36\), find \(y\) if \(x=4\).

    Answer

    \(y=64\)

    2)