Chapter 3

Find all zeros of the polynomial. $$P(x)=x^{3}-x-6$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\). $$P(x)=x^{4}+2 x^{3}-2 x^{2}-3 x+2$$

A quadratic function is given. (a) Use a graphing device to find the maximum or minimum value of the quadratic function \(f\) correct to two decimal places. (b) Find the exact maximum or minimum value of \(f,\) and compare it with your answer to part (a). $$f(x)=1+x-\sqrt{2} x^{2}$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=x^{3}-x+1, \quad c=\frac{1}{4}$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$(3-\sqrt{-5})(1+\sqrt{-1})$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$t(x)=\frac{3 x+6}{x^{2}+2 x-8}$$

Find all zeros of the polynomial. $$P(x)=2 x^{3}+7 x^{2}+12 x+9$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=x^{4}-7 x^{3}+14 x^{2}-3 x-9$$

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$y=-x^{2}+8 x, \quad[-4,12] \text { by }[-50,30]$$

Use synthetic division and the Remainder Theorem to evaluate \(P(c)\). $$P(x)=x^{3}+2 x^{2}-3 x-8, \quad c=0.1$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$(\sqrt{3}-\sqrt{-4})(\sqrt{6}-\sqrt{-8})$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$t(x)=\frac{x-2}{x^{2}-4 x}$$

Find all zeros of the polynomial. $$P(x)=2 x^{3}-8 x^{2}+9 x-9$$

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$y=x^{3}-3 x^{2}, \quad[-2,5] \text { by }[-10,10]$$

Let $$\begin{aligned}P(x)=& 6 x^{7}-40 x^{6}+16 x^{5}-200 x^{4} \\\&-60 x^{3}-69 x^{2}+13 x-139\end{aligned}$$ Calculate \(P(7)\) by (a) using synthetic division and (b) substituting \(x=7\) into the polynomial and evaluating directly.

Evaluate the radical expression and express the result in the form \(a+b i\) $$\frac{2+\sqrt{-8}}{1+\sqrt{-2}}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{(x-1)(x+2)}{(x+1)(x-3)}$$

Find all zeros of the polynomial. $$P(x)=x^{4}+x^{3}+7 x^{2}+9 x-18$$

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$y=x^{3}-12 x+9, \quad[-5,5] \text { by }[-30,30]$$

Use the Factor Theorem to show that \(x-c\) is a factor of \(P(x)\) for the given value(s) of \(c\). $$P(x)=x^{3}-3 x^{2}+3 x-1, \quad c=1$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$\frac{1-\sqrt{-1}}{1+\sqrt{-1}}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{2 x(x+2)}{(x-1)(x-4)}$$

Find all zeros of the polynomial. $$P(x)=x^{4}-2 x^{3}-2 x^{2}-2 x-3$$

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$y=2 x^{3}-3 x^{2}-12 x-32, \quad[-5,5] \text { by }[-60,30]$$

Use the Factor Theorem to show that \(x-c\) is a factor of \(P(x)\) for the given value(s) of \(c\). $$P(x)=x^{3}+2 x^{2}-3 x-10, \quad c=2$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$\frac{\sqrt{-36}}{\sqrt{-2} \sqrt{-9}}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{x^{2}-2 x+1}{x^{2}+2 x+1}$$

Find all zeros of the polynomial. $$P(x)=x^{5}-x^{4}+7 x^{3}-7 x^{2}+12 x-12$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=2 x^{4}+15 x^{3}+17 x^{2}+3 x-1$$

Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$f(x)=x^{3}-x$$

Use the Factor Theorem to show that \(x-c\) is a factor of \(P(x)\) for the given value(s) of \(c\). $$P(x)=2 x^{3}+7 x^{2}+6 x-5, \quad c=\frac{1}{2}$$

Evaluate the radical expression and express the result in the form \(a+b i\) $$\frac{\sqrt{-7} \sqrt{-49}}{\sqrt{28}}$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{4 x^{2}}{x^{2}-2 x-3}$$

Find all zeros of the polynomial. $$P(x)=x^{5}+x^{3}+8 x^{2}+8$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=4 x^{5}-18 x^{4}-6 x^{3}+91 x^{2}-60 x+9$$

Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$f(x)=3+x+x^{2}-x^{3}$$

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$y=x^{4}-18 x^{2}+32, \quad[-5,5] \text { by }[-100,100]$$

Use the Factor Theorem to show that \(x-c\) is a factor of \(P(x)\) for the given value(s) of \(c\). $$P(x)=x^{4}+3 x^{3}-16 x^{2}-27 x+63, \quad c=3,-3$$

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}+49=0$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{2 x^{2}+10 x-12}{x^{2}+x-6}$$

Find all zeros of the polynomial. $$P(x)=x^{4}-6 x^{3}+13 x^{2}-24 x+36$$

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\). (b) Sketch the graph of \(P\). $$P(x)=x^{3}-3 x^{2}-4 x+12$$

Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$g(x)=x^{4}-2 x^{3}-11 x^{2}$$

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$y=3 x^{5}-5 x^{3}+3, \quad[-3,3] \text { by }[-5,10]$$

Show that the given value(s) of \(c\) are zeros of \(P(x)\), and find all other zeros of \(P(x)\). $$P(x)=x^{3}-x^{2}-11 x+15, \quad c=3$$

Find all solutions of the equation and express them in the form \(a+b i\) $$9 x^{2}+4=0$$

Find the intercepts and asymptotes, and then sketch a graph of the rational function and state the domain and range. Use a graphing device to confirm your answer. $$r(x)=\frac{2 x^{2}+2 x-4}{x^{2}+x}$$

Find all zeros of the polynomial. $$P(x)=x^{4}-x^{2}+2 x+2$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=-x^{3}-2 x^{2}+5 x+6$$

Find the local maximum and minimum values of the function and the value of \(x\) at which each occurs. State each answer correct to two decimal places. $$g(x)=x^{5}-8 x^{3}+20 x$$