Chapter 4

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

\(41-58=\) Find all zeros of the polynomial. $$ P(x)=x^{5}-2 x^{4}+2 x^{3}-4 x^{2}+x-2 $$

\(51-58=\) 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 slant asymptote, the vertical asymptotes, and sketch a graph of the function. \(r(x)=\frac{x^{2}}{x-2}\)

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

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

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

\(59-64=\) 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 the slant asymptote, the vertical asymptotes, and sketch a graph of the function. \(r(x)=\frac{x^{2}-2 x-8}{x}\)

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

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

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

\(59-64=\) 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 the slant asymptote, the vertical asymptotes, and sketch a graph of the function. \(r(x)=\frac{3 x-x^{2}}{2 x-2}\)

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

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

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

\(59-64=\) 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. $$ P(x)=x^{4}+8 x^{2}-9 $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. \(r(x)=\frac{x^{2}+5 x+4}{x-3}\)

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

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

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

\(59-64=\) 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. $$ P(x)=x^{4}+8 x^{2}+16 $$

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

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

\(59-64=\) 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. $$ P(x)=x^{6}-64 $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. \(r(x)=\frac{x^{3}+x^{2}}{x^{2}-4}\)

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

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. $$ P(x)=x^{8}-x^{5}+x^{4}-x^{3}+x^{2}-x+1 $$

\(59-64=\) 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. $$ P(x)=x^{5}-16 x $$

Find the slant asymptote, the vertical asymptotes, and sketch a graph of the function. \(r(x)=\frac{2 x^{3}+2 x}{x^{2}-1}\)

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

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=2 x^{3}+5 x^{2}+x-2 ; \quad a=-3, b=1 $$

By the Zeros Theorem, every \(n\) th-degree polynomial equation has exactly \(n\) solutions (including possibly some that are repeated). Some of these may be real and some may be imaginary. Use a graphing device to determine how many real and imaginary solutions each equation has. (a) \(x^{4}-2 x^{3}-11 x^{2}+12 x=0\) (b) \(x^{4}-2 x^{3}-11 x^{2}+12 x-5=0\) (c) \(x^{4}-2 x^{3}-11 x^{2}+12 x+40=0\)

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. \(f(x)=\frac{2 x^{2}+6 x+6}{x+3}, \quad g(x)=2 x\)

Graph the family of polynomials in the same viewing rectangle, using the given values of \(c .\) Explain how changing the value of \(c\) affects the graph. $$ P(x)=c x^{3} ; \quad c=1,2,5, \frac{1}{2} $$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=x^{4}-2 x^{3}-9 x^{2}+2 x+8 ; \quad a=-3, b=5 $$

\(66-68=\) So far we have worked only with polynomials that have real coefficients. These exercises involve polynomials with real and imaginary coefficients. Find all solutions of the equation. (a) \(2 x+4 i=1\) (b) \(x^{2}-i x=0\) (c) \(x^{2}+2 i x-1=0\) (d) \(i x^{2}-2 x+i=0\)

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. \(f(x)=\frac{-x^{3}+6 x^{2}-5}{x^{2}-2 x}, g(x)=-x+4\)

Graph the family of polynomials in the same viewing rectangle, using the given values of \(c .\) Explain how changing the value of \(c\) affects the graph. $$ P(x)=(x-c)^{4} ; \quad c=-1,0,1,2 $$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=8 x^{3}+10 x^{2}-39 x+9 ; \quad a=-3, b=2 $$

\(66-68=\) So far we have worked only with polynomials that have real coefficients. These exercises involve polynomials with real and imaginary coefficients. (a) Show that 2\(i\) and \(1-i\) are both solutions of the equation $$ x^{2}-(1+i) x+(2+2 i)=0 $$ but that their complex conjugates \(-2 i\) and \(1+i\) are not. (b) Explain why the result of part (a) does not violate the Conjugate Zeros Theorem.

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. \(f(x)=\frac{x^{3}-2 x^{2}+16}{x-2}, g(x)=x^{2}\)

Suppose you were asked to solve the following two problems on a test: A. Find the remainder when \(6 x^{1000}-17 x^{562}+12 x+26\) is divided by \(x+1\) B. Is \(x-1\) a factor of \(x^{567}-3 x^{400}+x^{9}+2 ?\) Obviously, it's impossible to solve these problems by dividing, because the polynomials are of such large degree. Use one or more of the theorems in this section to solve these problems without actually dividing.

Graph the family of polynomials in the same viewing rectangle, using the given values of \(c .\) Explain how changing the value of \(c\) affects the graph. $$ P(x)=x^{4}+c ; \quad c=-1,0,1,2 $$

Show that the given values for \(a\) and \(b\) are lower and upper bounds for the real zeros of the polynomial. $$ P(x)=3 x^{4}-17 x^{3}+24 x^{2}-9 x+1 ; \quad a=0, b=6 $$

\(66-68=\) So far we have worked only with polynomials that have real coefficients. These exercises involve polynomials with real and imaginary coefficients. (a) Find the polynomial with real coefficients of the smallest possible degree for which \(i\) and \(1+i\) are zeros and in which the coefficient of the highest power is \(1 .\) (b) Find the polynomial with complex coefficients of the smallest possible degree for which \(i\) and \(1+i\) are zeros and in which the coefficient of the highest power is \(1 .\)

Graph the rational function \(f\) and determine all vertical asymptotes from your graph. Then graph \(f\) and \(g\) in a sufficiently large viewing rectangle to show that they have the same end behavior. \(f(x)=\frac{-x^{4}+2 x^{3}-2 x}{(x-1)^{2}}, \quad g(x)=1-x^{2}\)

Expand \(Q\) to prove that the polynomials \(P\) and \(Q\) are the same. $$\begin{array}{l}{P(x)=3 x^{4}-5 x^{3}+x^{2}-3 x+5} \\ {Q(x)=(((3 x-5) x+1) x-3) x+5}\end{array}$$ Try to evaluate \(P(2)\) and \(Q(2)\) in your head, using the forms given. Which is easier? Now write the polynomial \(R(x)=x^{5}-2 x^{4}+3 x^{3}-2 x^{2}+3 x+4\) in "nested" form, like the polynomial \(Q .\) Use the nested form to find \(R(3)\) in your head. Do you see how calculating with the nested form follows the same arithmetic steps as calculating the value of a polynomial using synthetic division?

Graph the family of polynomials in the same viewing rectangle, using the given values of \(c .\) Explain how changing the value of \(c\) affects the graph. $$ P(x)=x^{3}+c x ; \quad c=2,0,-2,-4 $$