Chapter 9

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(3 x^{4}-x^{3}+2 x^{2}-7 x-1\right) \div(x+1) $$

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. $$ f(x)=x^{3}+2 x^{2}-x-2 $$

For Problems 27-30, solve each equation by first applying the multiplication property of equality to produce an equivalent equation with integral coefficients. $$ \frac{1}{10} x^{3}+\frac{1}{5} x^{2}-\frac{1}{2} x-\frac{3}{5}=0 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x+2 \text { a factor of } x^{3}+7 x^{2}+x-18 ? $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(x^{4}+5 x^{3}-x^{2}+25\right) \div(x+5) $$

Each of the following graphs is a transformation of \(f(x)=\frac{1}{x}\). First predict the general shape and location of the graph, and then check your prediction with a graphing calculator. (a) \(f(x)=\frac{1}{x}-2\) (b) \(f(x)=\frac{1}{x+3}\) (c) \(f(x)=-\frac{1}{x}\) (d) \(f(x)=\frac{1}{x-2}+3\) (e) \(f(x)=\frac{2 x+1}{x}\)

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

For Problems 27-30, solve each equation by first applying the multiplication property of equality to produce an equivalent equation with integral coefficients. $$ \frac{1}{10} x^{3}+\frac{1}{2} x^{2}+\frac{1}{5} x-\frac{4}{5}=0 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x+3 \text { a factor of } x^{3}+x^{2}-14 x-24 ? $$

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

Graph \(f(x)=\frac{1}{x^{2}}\). How should the graph of \(f(x)=\) \(\frac{1}{(x-4)^{2}}, \quad f(x)=\frac{1+3 x^{2}}{x^{2}}\), and \(f(x)=\frac{1}{x^{2}}\) compare to the graph of \(f(x)=\frac{1}{x^{2}}\) ? Graph the three functions on the same set of axes with the graph of $$ f(x)=\frac{1}{x^{2}} \text {. } $$

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. $$ f(x)=x^{3}-8 x^{2}+19 x-12 $$

For Problems 27-30, solve each equation by first applying the multiplication property of equality to produce an equivalent equation with integral coefficients. $$ x^{3}-\frac{5}{6} x^{2}-\frac{22}{3} x+\frac{5}{2}=0 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-3 \text { a factor of } 3 x^{3}-5 x^{2}-17 x+17 ? $$

Graph \(f(x)=\frac{1}{x^{3}}\). How should the graphs of \(f(x)=\) \(\frac{2 x^{3}+1}{x^{3}}, f(x)=\frac{1}{(x+2)^{3}}\), and \(f(x)=\frac{-1}{x^{3}}\) compare to the graph of \(f(x)=\frac{1}{x^{3}}\) ? Graph the three functions on the same set of axes with the graph of \(f(x)=\frac{1}{x^{3}}\).

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. $$ f(x)=x^{3}+6 x^{2}+11 x+6 $$

For Problems 27-30, solve each equation by first applying the multiplication property of equality to produce an equivalent equation with integral coefficients. $$ x^{3}+\frac{9}{2} x^{2}-x-12=0 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x+4 \text { a factor of } 2 x^{3}+9 x^{2}-5 x-39 ? $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(x^{4}-16\right) \div(x+2) $$

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. $$ f(x)=2 x^{3}-3 x^{2}-3 x+2 $$

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

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

Suppose that \(x\) ounces of pure acid have been added to 14 ounces of a \(15 \%\) acid solution. (a) Set up the rational expression that represents the concentration of pure acid in the final solution. (b) Graph the rational function that displays the concentration. (c) How many ounces of pure acid need to be added to the 14 ounces of a \(15 \%\) solution to raise it to a \(40.5 \%\) solution? Check your answer. (d) How many ounces of pure acid need to be added to the 14 ounces of a \(15 \%\) solution to raise it to a \(50 \%\) solution? Check your answer. (e) What concentration of acid do we obtain if we add 12 ounces of pure acid to the 14 ounces of a \(15 \%\) solution? Check your answer.

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. $$ f(x)=x^{3}+2 x^{2}-x-2 $$

For Problems \(31-40\), use Descartes' rule of signs (Property 9.6) to help list the possibilities for the nature of the solutions for each equation. \(D o\) not solve the equations. $$ 8 x^{2}-14 x+3=0 $$

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

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

Solve the following problem both algebraically and graphically: One solution contains \(50 \%\) alcohol, and another solution contains \(80 \%\) alcohol. How many liters of each solution should be mixed to produce \(10.5\) liters of a \(70 \%\) alcohol solution? Check your answer.

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. $$ f(x)=x^{4}-5 x^{2}+4 $$

For Problems \(31-40\), use Descartes' rule of signs (Property 9.6) to help list the possibilities for the nature of the solutions for each equation. \(D o\) not solve the equations. $$ 2 x^{3}+x-3=0 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x-3 \text { a factor of } x^{4}-81 ? $$

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

Graph each of the following functions. Be sure that you get a complete graph for each one. Sketch each graph on a sheet of paper, and keep them all handy as you study the next section. (a) \(f(x)=\frac{x^{2}}{x^{2}-x-2}\) (b) \(f(x)=\frac{x}{x^{2}-4}\) (c) \(f(x)=\frac{3 x}{x^{2}+1}\) (d) \(f(x)=\frac{x^{2}-1}{x-2}\)

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. $$ f(x)=-x^{4}+5 x^{2}-4 $$

For Problems \(21-34\), use the factor theorem to help answer some questions about factors. $$ \text { Is } x+3 \text { a factor of } x^{4}-81 ? $$

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

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=(x+3)(x-6)(8-x) $$

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x-2, \quad f(x)=x^{3}-6 x^{2}-13 x+42 $$

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

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=(x-5)(x+4)(x-3) $$

For Problems \(31-40\), use Descartes' rule of signs (Property 9.6) to help list the possibilities for the nature of the solutions for each equation. \(D o\) not solve the equations. $$ 4 x^{3}+5 x^{2}-6 x-2=0 $$

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x+1, \quad f(x)=x^{3}+6 x^{2}-31 x-36 $$

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

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=(x+3)^{4}(x-1)^{3} $$

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x+2, \quad f(x)=12 x^{3}+29 x^{2}+8 x-4 $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(4 x^{5}-6 x^{4}+2 x^{3}+2 x^{2}-5 x+2\right) \div(x-1) $$

For Problems \(35-42\), (a) find the \(y\) intercepts, (b) find the \(x\) intercepts, and (c) find the intervals of \(x\) where \(f(x)>0\) and those where \(f(x)<0\). Do not sketch the graphs. $$ f(x)=(x-4)^{2}(x+3)^{3} $$

For Problems \(31-40\), use Descartes' rule of signs (Property 9.6) to help list the possibilities for the nature of the solutions for each equation. \(D o\) not solve the equations. $$ 2 x^{5}+3 x^{3}-x+1=0 $$

For Problems \(35-44\), use synthetic division to show that \(g(x)\) is a factor of \(f(x)\), and complete the factorization of \(f(x)\). $$ g(x)=x-3, \quad f(x)=6 x^{3}-17 x^{2}-5 x+6 $$

Use synthetic division to determine the quotient and remainder for each problem. $$ \left(3 x^{5}-8 x^{4}+5 x^{3}+2 x^{2}-9 x+4\right) \div(x-2) $$