Chapter 9

For Problems \(1-20\), graph each rational function. Check first for symmetry, and identify the asymptotes. $$ f(x)=\frac{x^{2}+4}{x+2} $$

Graph each of the following rational functions: $$ f(x)=\frac{x+2}{x} $$

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=x(x-2)^{2}(x-1) $$

For Problems \(1-20\), use the rational root theorem and the factor theorem to help solve each equation. Be sure that the number of solutions for each equation agrees with Property \(9.3\), taking into account multiplicity of solutions. $$ 3 x^{4}-x^{3}-8 x^{2}+2 x+4=0 $$

For Problems \(11-20\), find \(f(c)\) either by using synthetic division and the remainder theorem or by evaluating \(f(c)\) directly. $$ f(n)=-2 n^{5}-9 n^{4}+7 n^{3}+14 n^{2}+19 n-38 \text { and } c=-5 $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(-2 x^{3}-3 x^{2}+4 x+5\right) \div(x+1) $$

For Problems \(1-20\), graph each rational function. Check first for symmetry, and identify the asymptotes. $$ f(x)=\frac{x^{2}+1}{1-x} $$

Graph each of the following rational functions: $$ f(x)=\frac{4 x^{2}}{x^{2}+1} $$

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=(x-2)(x-1)(x+1)(x+2) $$

For Problems \(1-20\), use the rational root theorem and the factor theorem to help solve each equation. Be sure that the number of solutions for each equation agrees with Property \(9.3\), taking into account multiplicity of solutions. $$ 4 x^{4}+12 x^{3}+x^{2}-12 x+4=0 $$

For Problems \(11-20\), find \(f(c)\) either by using synthetic division and the remainder theorem or by evaluating \(f(c)\) directly. $$ f(x)=-4 x^{4}-6 x^{2}+7 \text { and } c=4 $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(-3 x^{3}+x^{2}+2 x+2\right) \div(x+1) $$

For Problems \(1-20\), graph each rational function. Check first for symmetry, and identify the asymptotes. $$ f(x)=\frac{x^{3}+8}{x^{2}} $$

Graph each of the following rational functions: $$ f(x)=\frac{4}{x^{2}+2} $$

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=(x-1)^{2}(x+2) $$

For Problems \(1-20\), use the rational root theorem and the factor theorem to help solve each equation. Be sure that the number of solutions for each equation agrees with Property \(9.3\), taking into account multiplicity of solutions. $$ 2 x^{5}-5 x^{4}+x^{3}+x^{2}-x+6=0 $$

For Problems \(11-20\), find \(f(c)\) either by using synthetic division and the remainder theorem or by evaluating \(f(c)\) directly. $$ f(x)=3 x^{5}-7 x^{3}-6 \text { and } c=5 $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(-x^{3}+4 x^{2}+31 x+2\right) \div(x-8) $$

Explain the concept of an oblique asymptote.

Graph each of the following rational functions: $$ f(x)=\frac{x^{2}-4}{x^{2}} $$

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=x(x-2)^{2}(x+1) $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-2 \text { a factor of } 5 x^{2}-17 x+14 ? $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(3 x^{3}-2 x-5\right) \div(x-2) $$

Explain why it is possible for curves to intersect horizontal and oblique asymptotes but not to intersect vertical asymptotes.

Graph each of the following rational functions: $$ f(x)=\frac{2 x^{4}}{x^{4}+1} $$

For Problems \(1-22\), graph each of the polynomial functions. $$ f(x)=(x+1)^{2}(x-1)^{2} $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x+1 \text { a factor of } 3 x^{2}-5 x-8 ? $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(2 x^{3}-x-4\right) \div(x+3) $$

Give a step-by-step description of how you would go about graphing \(f(x)=\frac{x^{2}-x-12}{x-2}\).

How would you explain the concept of an asymptote to an elementary algebra student?

For Problems \(23-34\), graph each polynomial function by first factoring the given polynomial. You may need to use some factoring techniques from Chapter 3 as well as the rational root theorem and the factor theorem. $$ f(x)=-x^{3}-x^{2}+6 x $$

For Problems \(21-26\), verify that the equations do not have any rational number solutions. $$ 3 x^{4}-4 x^{3}-10 x^{2}+3 x-4=0 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x+3 \text { a factor of } 6 x^{2}+13 x-14 ? $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(2 x^{4}+x^{3}+3 x^{2}+2 x-2\right) \div(x+1) $$

Give a step-by-step description of how you would go about graphing \(f(x)=\frac{-2}{x^{2}-9}\).

For Problems \(23-34\), graph each polynomial function by first factoring the given polynomial. You may need to use some factoring techniques from Chapter 3 as well as the rational root theorem and the factor theorem. $$ f(x)=x^{3}+x^{2}-2 x $$

For Problems \(21-26\), verify that the equations do not have any rational number solutions. $$ 2 x^{4}-3 x^{3}+6 x^{2}-24 x+5=0 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-5 \text { a factor of } 8 x^{2}-47 x+32 ? $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(x^{4}-3 x^{3}-6 x^{2}+11 x-12\right) \div(x-4) $$

First check for symmetry and identify the asymptotes for the graphs of the following rational functions. Then use your graphing utility to graph each function. (a) \(f(x)=\frac{4 x^{2}}{x^{2}+x-2}\) (b) \(f(x)=\frac{-2 x}{x^{2}-5 x-6}\) (c) \(f(x)=\frac{x^{2}}{x^{2}-9}\) (d) \(f(x)=\frac{x^{2}-4}{x^{2}-9}\) (e) \(f(x)=\frac{x^{2}-9}{x^{2}-4}\) (f) \(f(x)=\frac{x^{2}+2 x+1}{x^{2}-5 x+6}\)

The rational function \(f(x)=\frac{(x-2)(x+3)}{x-2}\) has a domain of all real numbers except 2 and can be simplified to \(f(x)=x+3\). Thus its graph is a straight line with a hole at \((2,5)\). Graph each of the following functions. See answer section. (a) \(f(x)=\frac{(x+4)(x-1)}{x+4}\) (b) \(f(x)=\frac{x^{2}-5 x+6}{x-2}\) (c) \(f(x)=\frac{x-1}{x^{2}-1}\) (d) \(f(x)=\frac{x+2}{x^{2}+6 x+8}\)

For Problems \(23-34\), graph each polynomial function by first factoring the given polynomial. You may need to use some factoring techniques from Chapter 3 as well as the rational root theorem and the factor theorem. $$ f(x)=x^{4}-5 x^{3}+6 x^{2} $$

For Problems \(21-26\), verify that the equations do not have any rational number solutions. $$ x^{5}+2 x^{4}-2 x^{3}+5 x^{2}-2 x-3=0 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-1 \text { a factor of } 4 x^{3}-13 x^{2}+21 x-12 \text { ? } $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(x^{4}+4 x^{3}-7 x-1\right) \div(x-3) $$

For each of the following rational functions, first determine and graph any oblique asymptotes. Then, on the same set of axes, graph the function. (a) \(f(x)=\frac{x^{2}-1}{x-2}\) (b) \(f(x)=\frac{x^{2}+1}{x+2}\) (c) \(f(x)=\frac{2 x^{2}+x+1}{x+1}\) (d) \(f(x)=\frac{x^{2}+4}{x-3}\) (e) \(f(x)=\frac{3 x^{2}-x-2}{x-2}\) (f) \(f(x)=\frac{4 x^{2}+x+1}{x+1}\) (g) \(f(x)=\frac{x^{3}+x^{2}-x-1}{x^{2}+2 x+3}\) (h) \(f(x)=\frac{x^{3}+2 x^{2}+x-3}{x^{2}-4}\)

Graph the function \(f(x)=x+2+\frac{3}{x-2}\). It may be necessary to plot a rather large number of points. Also, defend the statement that \(f(x)=x+2\) is an oblique asymptote. See answer section.

For Problems \(23-34\), graph each polynomial function by first factoring the given polynomial. You may need to use some factoring techniques from Chapter 3 as well as the rational root theorem and the factor theorem. $$ f(x)=-x^{4}-3 x^{3}-2 x^{2} $$

For Problems \(21-26\), verify that the equations do not have any rational number solutions. $$ x^{5}-2 x^{4}+3 x^{3}+4 x^{2}+7 x-1=0 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-4 \text { a factor of } 2 x^{3}-11 x^{2}+10 x+8 ? $$