Chapter 3

Graph the polynomial in the given viewing rectangle. Find the coordinates of all local extrema. State each answer rounded to two decimal places. $$y=x^{5}-5 x^{2}+6, \quad[-3,3] \text { by }[-5,10]$$

Show that the given value(s) of \(c\) are zeros of \(P(x)\), and find all other zeros of \(P(x)\). $$P(x)=3 x^{4}-x^{3}-21 x^{2}-11 x+6, \quad c=\frac{1}{3},-2$$

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}-4 x+5=0$$

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. $$r(x)=\frac{x^{2}-x-6}{x^{2}+3 x}$$

Find all zeros of the polynomial. $$P(x)=4 x^{4}+4 x^{3}+5 x^{2}+4 x+1$$

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

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. $$U(x)=x \sqrt{6-x}$$

Graph the polynomial and determine how many local maxima and minima it has. $$y=-2 x^{2}+3 x+5$$

Find a polynomial of the specified degree that has the given zeros. Degree \(3 ; \quad\) zeros -1,1,3

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

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. $$r(x)=\frac{x^{2}+3 x}{x^{2}-x-6}$$

Find all zeros of the polynomial. $$P(x)=4 x^{4}+2 x^{3}-2 x^{2}-3 x-1$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=3 x^{3}+17 x^{2}+21 x-9$$

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. $$U(x)=x \sqrt{x-x^{2}}$$

Graph the polynomial and determine how many local maxima and minima it has. $$y=x^{3}+12 x$$

Find a polynomial of the specified degree that has the given zeros. Degree \(4 ; \quad\) zeros -2,0,2,4

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}+2 x+5=0$$

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. $$r(x)=\frac{3 x^{2}+6}{x^{2}-2 x-3}$$

Find all zeros of the polynomial. $$P(x)=x^{5}-3 x^{4}+12 x^{3}-28 x^{2}+27 x-9$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=x^{4}-5 x^{3}+6 x^{2}+4 x-8$$

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. $$v(x)=\frac{1-x^{2}}{x^{3}}$$

Graph the polynomial and determine how many local maxima and minima it has. $$y=x^{3}-x^{2}-x$$

Find a polynomial of the specified degree that has the given zeros. Degree \(4 ; \quad\) zeros -1,1,3,5

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}-6 x+10=0$$

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. $$r(x)=\frac{5 x^{2}+5}{x^{2}+4 x+4}$$

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

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=-x^{4}+10 x^{2}+8 x-8$$

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. $$V(x)=\frac{1}{x^{2}+x+1}$$

Graph the polynomial and determine how many local maxima and minima it has. $$y=6 x^{3}+3 x+1$$

Find a polynomial of the specified degree that has the given zeros. Degree \(5 ; \quad\) zeros -2,-1,0,1,2

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}+x+1=0$$

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. $$s(x)=\frac{x^{2}-2 x+1}{x^{3}-3 x^{2}}$$

A polynomial \(P\) is given. (a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. $$P(x)=x^{3}-5 x^{2}+4 x-20$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=x^{5}-x^{4}-5 x^{3}+x^{2}+8 x+4$$

If a ball is thrown directly upward with a velocity of \(40 \mathrm{ft} / \mathrm{s}\), its height (in feet) after \(t\) seconds is given by \(y=40 t-16 t^{2} .\) What is the maximum height attained by the ball?

Graph the polynomial and determine how many local maxima and minima it has. $$y=x^{4}-5 x^{2}+4$$

Find a polynomial of degree 3 that has zeros \(1,-2,\) and 3 and in which the coefficient of \(x^{2}\) is \(3 .\)

Find all solutions of the equation and express them in the form \(a+b i\) $$x^{2}-3 x+3=0$$

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. $$t(x)=\frac{x^{3}-x^{2}}{x^{3}-3 x-2}$$

A polynomial \(P\) is given. (a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. $$P(x)=x^{3}-2 x-4$$

Find all the real zeros of the polynomial. Use the quadratic formula if necessary, as in Example \(3(a)\) $$P(x)=x^{5}-x^{4}-6 x^{3}+14 x^{2}-11 x+3$$

A ball is thrown across a playing field from a height of \(5 \mathrm{ft}\) above the ground at an angle of \(45^{\circ}\) to the horizontal at a speed of \(20 \mathrm{ft} / \mathrm{s}\). It can be deduced from physical principles that the path of the ball is modeled by the function $$y=-\frac{32}{(20)^{2}} x^{2}+x+5$$ where \(x\) is the distance in feet that the ball has traveled horizontally. (a) Find the maximum height attained by the ball. (b) Find the horizontal distance the ball has traveled when it hits the ground.

Graph the polynomial and determine how many local maxima and minima it has. $$y=1.2 x^{5}+3.75 x^{4}-7 x^{3}-15 x^{2}+18 x$$

Find all solutions of the equation and express them in the form \(a+b i\) $$2 x^{2}-2 x+1=0$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{2}}{x-2}$$

A polynomial \(P\) is given. (a) Factor \(P\) into linear and irreducible quadratic factors with real coefficients. (b) Factor \(P\) completely into linear factors with complex coefficients. $$P(x)=x^{4}+8 x^{2}-9$$

\(65-70\) - Use Descartes' Rule of Signs to determine how many positive and how many negative real zeros the polynomial can have. Then determine the possible total number of real zeros. $$P(x)=x^{3}-x^{2}-x-3$$

A manufacturer finds that the revenue generated by selling \(x\) units of a certain commodity is given by the function \(R(x)=80 x-0.4 x^{2},\) where the revenue \(R(x)\) is measured in dollars. What is the maximum revenue, and how many units should be manufactured to obtain this maximum?

Find all solutions of the equation and express them in the form \(a+b i\) $$2 x^{2}+3=2 x$$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$r(x)=\frac{x^{2}+2 x}{x-1}$$