Chapter 4

Find all real solutions. Do not use a calculator. $$x^{3}-25 x=0$$

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{10 x^{6}}{5 x^{3}}$$

Find a cubic polynomial in standard form with real coefficients. having the given zeros. Let the leading coefficient be 1. Do not use a calculator. 4 and \(2+i\)

Find all real solutions. Do not use a calculator. $$x^{4}-x^{3}-6 x^{2}=0$$

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{6 x^{4}}{2 x^{3}}$$

Find a cubic polynomial in standard form with real coefficients. having the given zeros. Let the leading coefficient be 1. Do not use a calculator. \(-3\) and \(6+2 i\)

Find all real solutions. Do not use a calculator. $$x^{4}-x^{2}=2 x^{2}+4$$

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{8 x^{9}}{3 x^{7}}$$

Find a cubic polynomial in standard form with real coefficients. having the given zeros. Let the leading coefficient be 1. Do not use a calculator. 5 and \(i\)

Find all real solutions. Do not use a calculator. $$x^{4}+5=6 x^{2}$$

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$-\frac{2 x^{5}}{7 x^{2}}$$

Find a cubic polynomial in standard form with real coefficients. having the given zeros. Let the leading coefficient be 1. Do not use a calculator. \(-9\) and \(-i\)

Find all real solutions. Do not use a calculator. $$x^{3}-3 x^{2}-18 x=0$$

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{2 x^{6}+3 x^{3}}{2 x}$$

Find a cubic polynomial in standard form with real coefficients. having the given zeros. Let the leading coefficient be 1. Do not use a calculator. 0 and \(3+i\)

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{5 x^{3}+x^{2}}{3 x^{2}}$$

Find all real solutions. Do not use a calculator. $$x^{4}-x^{2}=0$$

Find a cubic polynomial in standard form with real coefficients. having the given zeros. Let the leading coefficient be 1. Do not use a calculator. 0 and \(4-3 i\)

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of \(-3,-1,\) and \(4 ; \quad P(2)=5\)

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{8 x^{3}-5 x}{2 x}$$

Find all real solutions. Do not use a calculator. $$2 x^{3}=4 x^{2}-2 x$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of \(1,-1,\) and \(0 ; \quad P(2)=-3\)

Simplify each rational expression. Apply the property \(\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}\) if necessary. Do not use a calculator. $$\frac{7 x^{8}-6 x^{3}}{6 x^{2}}$$

Find all real solutions. Do not use a calculator. $$x^{3}=x$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=\sqrt{5} x^{3}+2 x^{2}-3 x+4$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of \(-2,1,\) and \(0 ; \quad P(-1)=-1\)

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. $$P(x)=3 x^{2}-2 x-6 ; \quad 1 \text { and } 2$$

Find all real solutions. Do not use a calculator. $$12 x^{3}=17 x^{2}+5 x$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=-\sqrt{7} x^{3}-4 x^{2}+2 x-1$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of \(2,5,\) and \(-3 ; \quad P(1)=-4\)

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. $$P(x)=-x^{3}-x^{2}+5 x+5 ; 2 \text { and } 3$$

Find all real solutions. Do not use a calculator. $$3 x^{3}+3 x=10 x^{2}$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=-\pi x^{5}+3 x^{2}-1$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of 4 and \(1+i ; P(2)=4\)

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. $$P(x)=2 x^{3}-8 x^{2}+x+16 ; 2 \text { and } 2.5$$

Find all real solutions. Do not use a calculator. $$2 x^{3}+4=x(x+8)$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=\pi x^{7}-x^{5}+x-1$$

Find a polynomial function \(P(x)\) of degree 3 with real coefficients that satisfies the given conditions. Do not use a calculator. Zeros of \(-7\) and \(2-i ; \quad P(1)=9\)

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. $$P(x)=3 x^{3}+7 x^{2}-4 ; \quad 0.5 \text { and } 1$$

Find all real solutions. Do not use a calculator. $$3 x^{3}+18=x(2 x+27)$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=2.74 x^{4}-3 x^{2}+x-2$$

One or more zeros are given for each polynomial. Find all remaining zeros. \(P(x)=x^{3}-x^{2}-4 x-6 ; \quad 3\) 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. $$P(x)=2 x^{4}-4 x^{2}+3 x-6 ; \quad 1.5 \text { and } 2$$

Find all complex solutions of each equation. Do not use a calculator. $$7 x^{3}+x=0$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=\sqrt{6} x^{6}-x^{5}+2 x-2$$

One or more zeros are given for each polynomial. Find all remaining zeros. \(P(x)=x^{3}-5 x^{2}+17 x-13 ; \quad 1\) 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. $$P(x)=x^{4}-4 x^{3}-x+1 ; \quad 0.3 \text { and } 1$$

Find all complex solutions of each equation. Do not use a calculator. $$2 x^{3}+4 x=0$$

Describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=x^{5}-x^{4}-\pi x^{6}-x+3$$

One or more zeros are given for each polynomial. Find all remaining zeros. \(P(x)=3 x^{4}-2 x^{3}-26 x^{2}+18 x-9 ;-3\) and 3 are Zeros.