Chapter 12

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$z \bar{z}$$

Use long division to find the quotients and the remainders. Also, write each answer in the form \(p(x)=d(x) \cdot q(x)+R(x),\) as in equation (2) in the text. $$\frac{z^{5}-1}{z-1}$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. \frac{1}{x^{3}+x^{2}-10 x+8}

Express each polynomial in the form \(a_{n}\left(x-r_{1}\right)\left(x-r_{2}\right) \cdots\left(x-r_{n}\right)\). $$x^{3}+2 x^{2}-3 x-6$$

Find a quadratic equation with rational coefficients, one of whose roots is the given number. Write your answer so that the coefficient of \(x^{2}\) is 1. Use either of the methods shown in Example 3 $$r_{1}=2-\sqrt{3}$$

You are given a polynomial equation \(f(x)=0 .\) Specify the multiplicity of each repeated root. Then use a graphing utility to visually verify that the graph of \(y=f(x)\) is tangent to the \(x\) -axis at each repeated root. (a) \(x(x+1)^{2}(x-1)=0\) (b) \(x^{2}(x+1)^{2}(x-1)=0\) (c) \(x^{3}(x+1)^{2}(x-1)=0\)

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$w \bar{w}$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{5-x}{6 x^{2}-19 x+15}$$

Find a polynomial \(f(x)\) with leading coefficient 1 such that the equation \(f(x)=0\) has the given roots and no others. If the degree of \(f(x)\) is 7 or more, express \(f(x)\) in factored form; otherwise, express \(f(x)\) in the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\). $$\begin{array}{lcc}\hline \text { Root } & 1 & -3 \\\\\text { Multiplicity } & 2 & 1 \\\\\hline\end{array}$$

Find the rational roots of each equation, and then solve the equation. (Use the rational roots theorem and the upper and lower bound theorem, as in Example 2.) $$4 x^{3}+x^{2}-20 x-5=0$$

Find a quadratic equation with rational coefficients, one of whose roots is the given number. Write your answer so that the coefficient of \(x^{2}\) is 1. Use either of the methods shown in Example 3 $$r_{1}=(2+\sqrt{10}) / 3$$

Use the remainder theorem to evaluate \(f(x)\) for the given value of \(x\). $$f(x)=4 x^{3}-6 x^{2}+x-5 ; x=-3$$

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$z\left(w w_{1}\right)$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{2 x}{32 x^{2}-12 x+1}$$

Find a polynomial \(f(x)\) with leading coefficient 1 such that the equation \(f(x)=0\) has the given roots and no others. If the degree of \(f(x)\) is 7 or more, express \(f(x)\) in factored form; otherwise, express \(f(x)\) in the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\). $$\begin{array}{lll} \hline \text { Root } & 0 & 4 \\ \text { Multiplicity } & 2 & 1 \\ \hline \end{array}$$

Find the rational roots of each equation, and then solve the equation. (Use the rational roots theorem and the upper and lower bound theorem, as in Example 2.) $$3 x^{3}-16 x^{2}+17 x-4=0$$

Find a quadratic equation with rational coefficients, one of whose roots is the given number. Write your answer so that the coefficient of \(x^{2}\) is 1. Use either of the methods shown in Example 3 $$r_{1}=\frac{1}{2}+\frac{1}{4} \sqrt{5}$$

Use the remainder theorem to evaluate \(f(x)\) for the given value of \(x\). $$f(x)=2 x^{3}-x-4 ; x=4$$

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$(z w) w_{1}$$

\(Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \)17,18,\( and \)26,\( use the rational roots theorem to factor the denominator. \)\frac{2 x+1}{x^{3}-5 x}$$

Find a polynomial \(f(x)\) with leading coefficient 1 such that the equation \(f(x)=0\) has the given roots and no others. If the degree of \(f(x)\) is 7 or more, express \(f(x)\) in factored form; otherwise, express \(f(x)\) in the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\). $$\begin{array}{lllll} \hline \text { Root } & 2 & -2 & 2 i & -2 i \\ \text { Multiplicity } & 1 & 1 & 1 & 1 \\ \hline \end{array}$$

Let \(f(x)=2 x^{4}-3 x^{3}+12 x^{2}+22 x-60\) (a) Use Descartes's rule to verify that the equation \(f(x)=0\) has one negative root. (b) Use Descartes's rule to verify that the equation \(f(x)=0\) has either one or three positive roots. (c) Graph the equation \(y=f(x) .\) Use the graph to say which of the two cases in part (b) actually holds. (d) Use the graph to estimate the real roots of the equation \(f(x)=0 .\) Check that your answers are consistent with the values obtained in Example \(1 .\)

Use the remainder theorem to evaluate \(f(x)\) for the given value of \(x\). $$f(x)=6 x^{4}+5 x^{3}-8 x^{2}-10 x-3 ; x=1 / 2$$

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$z\left(w+w_{1}\right)$$

Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form \(p(x)=d(x) \cdot q(x)+R(x),\) as in equation (2) in the text. $$\frac{x^{2}-6 x-2}{x-5}$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{2 x+1}{x^{3}+5 x}$$

Find a polynomial \(f(x)\) with leading coefficient 1 such that the equation \(f(x)=0\) has the given roots and no others. If the degree of \(f(x)\) is 7 or more, express \(f(x)\) in factored form; otherwise, express \(f(x)\) in the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\). $$\begin{array}{lcc} \hline \text { Root } & 2+i & 2-i \\ \text { Multiplicity } & 1 & 1 \\ \hline \end{array}$$

Find the rational roots of each equation, and then solve the equation. (Use the rational roots theorem and the upper and lower bound theorem, as in Example 2.) $$4 x^{3}-10 x^{2}-25 x+4=0$$

Use the remainder theorem to evaluate \(f(x)\) for the given value of \(x\). $$f(x)=x^{5}-x^{4}-x^{3}-x^{2}-x-1 ; x=-2$$

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$z w+z w_{1}$$

Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form \(p(x)=d(x) \cdot q(x)+R(x),\) as in equation (2) in the text. $$\frac{3 x^{2}+4 x-1}{x-1}$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{x^{3}+2}{x^{4}+8 x^{2}+16}$$

Find a polynomial \(f(x)\) with leading coefficient 1 such that the equation \(f(x)=0\) has the given roots and no others. If the degree of \(f(x)\) is 7 or more, express \(f(x)\) in factored form; otherwise, express \(f(x)\) in the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\). $$\begin{array}{lcccc} \hline \text { Root } & \sqrt{3} & -\sqrt{3} & 4 i & -4 i \\ \text { Multiplicity } & 2 & 2 & 1 & 1 \\ \hline \end{array}$$

(a) Find an appropriate viewing rectangle to demonstrate that the following purported partial fraction decomposition is incorrect: $$\frac{2 x+5}{(x-4)(x+3)}=\frac{13 / 7}{x-4}+\frac{2 / 7}{x+3}$$ (b) Follow part (a) using $$\frac{2 x+5}{(x-4)(x+3)}=\frac{13 / 7}{x-4}-\frac{1 / 7}{x+3}$$ (c) Determine the correct partial fraction decomposition. given that it has the general form $$\frac{2 x+5}{(x-4)(x+3)}=\frac{A}{x-4}+\frac{B}{x+3}$$

Use the remainder theorem to evaluate \(f(x)\) for the given value of \(x\). $$f(x)=x^{2}+3 x-4 ; x=-\sqrt{2}$$

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$z^{2}-w^{2}$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{x^{3}+2}{x^{4}-8 x^{2}+16}$$

Find a polynomial \(f(x)\) with leading coefficient 1 such that the equation \(f(x)=0\) has the given roots and no others. If the degree of \(f(x)\) is 7 or more, express \(f(x)\) in factored form; otherwise, express \(f(x)\) in the form \(a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0}\). $$\begin{array}{lcccc} \hline \text { Root } & 5 & 1 & 1-i & 1+i \\ \text { Multiplicity } & 2 & 3 & 1 & 1 \\ \hline \end{array}$$

(a) Find an appropriate viewing rectangle to demonstrate that the following purported partial fraction decomposition is incorrect: $$\frac{4}{x^{2}(x-5)}=\frac{-4 / 5}{x^{2}}+\frac{4 / 25}{x-5}$$ (b) Follow part (a) using $$\frac{4}{x^{2}(x-5)}=\frac{-3 / 25}{x}+\frac{-2 / 5}{x^{2}}+\frac{6 / 25}{x-5}$$ (c) Determine the correct partial fraction decomposition. given that it has the general form $$\frac{4}{x^{2}(x-5)}=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C}{x-5}$$

Let \(f(x)=x^{3}-x^{2}+3 x+2\) (a) Use Descartes's rule to explain in complete sentences why the equation \(f(x)=0\) has either: one negative root and two positive roots or: one negative root and two nonreal complex roots (If you need help, review Example 6 in the text.) (b) Use a graph to determine which of the two possibilities in part (a) is actually the case. (c) Use a graphing utility to compute the real \(\operatorname{root}(\mathrm{s})\) of the equation \(f(x)=0\)

Use the remainder theorem to evaluate \(f(x)\) for the given value of \(x\). $$f(x)=x^{7}-7 x^{6}+5 x^{4}+1 ; x=-3$$

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$(z-w)(z+w)$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{x^{3}+x-3}{x^{4}-15 x^{3}+75 x^{2}-125 x}$$

(a) Find a polynomial \(f(x)\) with leading coefficient 1 such that the equation \(f(x)=0\) has the given roots and no others. If the degree of \(f(x)\) is more than \(4,\) leave \(f(x)\) in factored form rather than multiplying it out. (b) Use a graphing utility to check the following fact, mentioned in this section: If \(x=r\) is a multiple root of \(f(x)=0\), then the graph of the function f is tangent to the \(x\) -axis at \(x=r\). $$\begin{array}{lccc} \hline \text { Root } & 0 & 1 & 3 \\ \text { Multiplicity } & 2 & 1 & 1 \\ \hline \end{array}$$

(a) Solve the following system of equations. (As indicated in Example \(3,\) you should obtain \(A=3, B=4\) and \(C=-1 .)\) $$\left\\{\begin{aligned} A+B &=7 \\ -2 B+C &=-9 \\ 9 A-2 C &=29 \end{aligned}\right.$$ (b) Solve the following system of equations. (As indicated in the text, you should obtain \(B=4\) and \(C=-1 .)\) $$\left\\{\begin{array}{c} -9 B-2 C=-34 \\ -2 B+C=-9 \end{array}\right.$$

Use Descartes's rule of signs to obtain information regarding the roots of the equations. $$x^{3}+5=0$$

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$(z w)^{2}$$

Use synthetic division to find the quotients and remainders. Also, in each case, write the result of the division in the form \(p(x)=d(x) \cdot q(x)+R(x),\) as in equation (2) in the text. $$\frac{6 x^{3}-5 x^{2}+2 x+1}{x-4}$$

Determine the partial fraction decomposition for each of the given rational expressions. Hint: In Exercises \(17,18,\) and \(26,\) use the rational roots theorem to factor the denominator. $$\frac{4 x^{2}}{2 x^{3}-5 x^{2}-4 x+3}$$

(a) Find a polynomial \(f(x)\) with leading coefficient 1 such that the equation \(f(x)=0\) has the given roots and no others. If the degree of \(f(x)\) is more than \(4,\) leave \(f(x)\) in factored form rather than multiplying it out. (b) Use a graphing utility to check the following fact, mentioned in this section: If \(x=r\) is a multiple root of \(f(x)=0\), then the graph of the function f is tangent to the \(x\) -axis at \(x=r\). $$\begin{array}{llll} \hline \text { Root } & 0 & 1 & 3 \\ \text { Multiplicity } & 1 & 2 & 1 \\ \hline \end{array}$$