Chapter 4

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use a calculator to approximate the zero to the nearest hundredth. $$P(x)=-x^{4}+2 x^{3}+x+12 ; \quad 2.7 \text { and } 2.8$$

Find all complex solutions of each equation. Do not use a calculator. $$3 x^{3}+2 x^{2}-3 x-2=0$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=-x-3.2 x^{3}+x^{2}-2.84 x^{4}$$

One or more zeros are given for each polynomial. Find all remaining zeros. \(P(x)=2 x^{4}-x^{3}-27 x^{2}+16 x-80 ;-4\) and 4 are Zeros.

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use a calculator to approximate the zero to the nearest hundredth. $$P(x)=-2 x^{4}+x^{3}-x^{2}+3 ;-1 \text { and }-0.9$$

Find all complex solutions of each equation. Do not use a calculator. $$x^{4}-11 x^{2}+10=0$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=x^{10,000}$$

One or more zeros are given for each polynomial. Find all remaining zeros. P(x)=x^{4}-x^{3}+10 x^{2}-9 x+9 ; \quad 3 i \text { is a zero. }

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use a calculator to approximate the zero to the nearest hundredth. $$P(x)=x^{5}-2 x^{3}+1 ;-1.6 \text { and }-1.5$$

Find all complex solutions of each equation. Do not use a calculator. $$5 x^{3}-x^{2}+10 x-2=0$$

One or more zeros are given for each polynomial. Find all remaining zeros. P(x)=2 x^{4}-2 x^{3}+55 x^{2}-50 x+125 ; \quad-5 i \text { is a zero. }

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use a calculator to approximate the zero to the nearest hundredth. $$P(x)=2 x^{7}-x^{4}+x-4 ; \quad 1.1 \text { and } 1.2$$

Find all complex solutions of each equation. Do not use a calculator. $$x^{4}+x^{2}-6=0$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=-3 x^{15,297}$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. 5 and \(-4\)

Suppose that a polynomial function \(P\) is defined in such a way that \(P(2)=-4\) and \(P(2.5)=2\) What conclusion does the intermediate value theorem allow you to make?

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&4 x^{4}-25 x^{2}+36=0\\\&[-5,5] \text { by }\lfloor- 5,100\rfloor\end{aligned}$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=12 x^{107,499}$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. 6 and \(-2\)

Suppose that a polynomial function \(P\) is defined in such a way that \(P(3)=-4\) and \(P(4)=-10 .\) Can we be certain that there is no zero between 3 and \(4 ?\) Explain, using a graph.

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&4 x^{4}-29 x^{2}+25=0\\\&[-5,5] \text { by }\lfloor- 50,100\rfloor\end{aligned}$$

Give a short written answer. The graphs of \(f(x)=x^{n}\) for \(n=3,5,7, \ldots\) resemble each other. As \(n\) gets larger, what happens to the graph?

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(-3,2,\) and \(i\)

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{3}+2 x^{2}-17 x-10 ; \quad x+5$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&x^{4}-15 x^{2}-16=0\\\&[-5,5] \text { by }\lfloor- 100,100\rfloor\end{aligned}$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(1+\sqrt{2}, 1-\sqrt{2},\) and 3

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{4}+4 x^{3}+2 x^{2}+9 x+4 ; \quad x+4$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&9 x^{4}+35 x^{2}-4=0\\\&[-3,3] \text { by }[-10,100]\end{aligned}$$

Give a short written answer. Using a window of \([-1,1]\) by \([-1,1],\) graph the odd degree polynomial functions $$y=x, \quad y=x^{3}, \quad \text { and } \quad y=x^{5}$$ Describe the behavior of these functions relative to each other. Predict the behavior of the graph of \(y=x^{7}\) in the same window, and then graph it to support your prediction.

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(1-\sqrt{3}, 1+\sqrt{3},\) and 1

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=3 x^{3}-11 x^{2}-20 x+3 ; \quad x-5$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&x^{3}-x^{2}-64 x+64=0\\\&[-10,10] \text { by }[-300,300]\end{aligned}$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(-2+i,-2-i, 3,\) and \(-3\)

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{4}-3 x^{3}-5 x^{2}+2 x-16 ; \quad x-3$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&x^{3}+6 x^{2}-100 x-600=0\\\&[-15,15] \text { by }[-1000,300]\end{aligned}$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(3+2 i,-1,\) and 2

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{4}-3 x^{3}-4 x^{2}+12 x ; \quad x-2$$

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&-2 x^{3}-x^{2}+3 x=0\\\&[-4,4] \text { by }[-10,10]\end{aligned}$$

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=2 x^{4}+3 x^{3}-5 x^{2}-18 x ; \quad x-2$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. 2 and \(3 i\)

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&-5 x^{3}+13 x^{2}+6 x=0\\\&[-4,4] \text { by }[-2,30]\end{aligned}$$

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{3}+2 x^{2}-3 ; x-1$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(6-3 i\) and \(-1\) (multiplicity 2 )

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&x^{3}+x^{2}-7 x-7=0\\\&[-10,10] \text { by }[-20,20]\end{aligned}$$

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=x^{3}-2 x^{2}-9 ; \quad x-3$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(1+2 i \text { and } 2 \text { (multiplicity } 2)\)

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&x^{3}+3 x^{2}-19 x-57=0\\\&[-10,10] \text { by }[-100,50]\end{aligned}$$

Find each quotient when \(P(x)\) is divided by the specified binomial. $$P(x)=-2 x^{3}-x-2 ; \quad x+1$$

Find a polynomial function \(P(x)\) having leading coefficient 1, least possible degree, real coefficients. and the given zeros. \(2+i \text { and }-3 \text { (multiplicity } 2)\)

Solve each equation analyrically for all complex solutions, giving exact forms in your solution set. Then graph the left side of the equation as \(Y_{1}\) in the suggested viewing window and, using the capabilities of vour calculater, suppert the real solutions. $$\begin{aligned}&-3 x^{3}-x^{2}+6 x=0\\\&[-4,4] \text { by }[-10,10]\end{aligned}$$