Chapter 4

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=-2 x^{2}+3 x+5 $$

\(59-62\) . Find a polynomial of the specified degree that has the given zeros. Degree \(3 ; \quad\) zeros \(-1,1,3\)

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

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

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=3 x^{3}+17 x^{2}+21 x-9 $$

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

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=x^{3}+12 x $$

\(59-62\) . Find a polynomial of the specified degree that has the given zeros. Degree \(4 ; \quad\) zeros \(-2,0,2,4\)

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

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

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=x^{4}-5 x^{3}+6 x^{2}+4 x-8 $$

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

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=x^{3}-x^{2}-x $$

\(59-62\) . Find a polynomial of the specified degree that has the given zeros. Degree \(4 ; \quad\) zeros \(-1,1,3,5\)

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

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=-x^{4}+10 x^{2}+8 x-8 $$

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

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=6 x^{3}+3 x+1 $$

\(59-62\) . Find a polynomial of the specified degree that has the given zeros. Degree \(5 ; \quad\) zeros \(-2,-1,0,1,2\)

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

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

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=x^{5}-x^{4}-5 x^{3}+x^{2}+8 x+4 $$

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

\(59-68\) 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 .

Height of a Ball 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?

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. \(P(x)=x^{3}-5 x^{2}+4 x-20\)

A polynomial \(P\) is given. (a) Find all the real zeros of \(P\) (b) Sketch the graph of \(P\) . $$ P(x)=x^{5}-x^{4}-6 x^{3}+14 x^{2}-11 x+3 $$

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

\(59-68\) 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 a polynomial of degree 4 that has integer coefficients and zeros \(1,-1,2,\) and \(\frac{1}{2} .\)

Path of a Ball 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.

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. \(P(x)=x^{3}-2 x-4\)

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

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

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=(x-2)^{5}+32 $$

Revenue 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?

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. \(P(x)=x^{4}+8 x^{2}-9\)

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. $$ P(x)=2 x^{3}-x^{2}+4 x-7 $$

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

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=\left(x^{2}-2\right)^{3} $$

Sales A soft-drink vendor at a popular beach analyzes his sales records and finds that if he sells \(X\) cans of soda pop in one day, his profit (in dollars) is given by $$ P(x)=-0.001 x^{2}+3 x-1800 $$ What is his maximum profit per day, and how many cans must he sell for maximum profit?

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. \(P(x)=x^{4}+8 x^{2}+16\)

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. $$ P(x)=2 x^{6}+5 x^{4}-x^{3}-5 x-1 $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{x^{2}-2 x-8}{x} $$

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=x^{8}-3 x^{4}+x $$

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. \(P(x)=x^{6}-64\)

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. $$ P(x)=x^{4}+x^{3}+x^{2}+x+12 $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. $$ r(x)=\frac{3 x-x^{2}}{2 x-2} $$

\(59-68\) Graph the polynomial and determine how many local maxima and minima it has. $$ y=\frac{1}{3} x^{7}-17 x^{2}+7 $$