Chapter 3

Find the domain and the vertical and horizontal asymptotes (if any). $$F(x)=\frac{4}{x-3}$$

For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$f(x)=x^{2}+9, x=-3,3$$

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$2 x^{4}-x^{3}+x^{2}-x ; 2 x+1$$

Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touch the \(x\) -axis? You may check your results with a graphing utility. $$g(s)=(s+6)^{4}(s-3)^{3}$$

Complete them to review topics relevant to the remaining exercises. Factor: \(2 x^{3}-50 x\)

Solve the polynomial inequality. $$x^{3}-16 x<0$$

For each polynomial function, list the zeros of the polynomial and state the multiplicity of each zero. $$f(s)=(s-\pi)^{10}(s+\pi)^{3}$$

Find the domain and the vertical and horizontal asymptotes (if any). $$g(x)=\frac{3}{x^{2}-4}$$

For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$f(x)=x^{3}+2 x^{2}-3 x-6 ; x=\sqrt{3},-\sqrt{2}$$

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$x^{6}+1 ; x+1$$

Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touch the \(x\) -axis? You may check your results with a graphing utility. $$h(t)=t^{2}(t-1)(t+2)$$

Complete them to review topics relevant to the remaining exercises. The graph of \(f(x)=x^{2}+3\) is the graph of \(y=x^{2}\) shifted __________ 3 units.

Solve the polynomial inequality. $$x^{3}-9 x>0$$

For each polynomial function, list the zeros of the polynomial and state the multiplicity of each zero. $$h(x)=(x-\sqrt{2})^{13}(x+\sqrt{2})^{7}$$

For each polynomial, determine which of the numbers listed next to it are zeros of the polynomial. $$(x)=x^{3}+2 x^{2}-2 x-4 ; x=\sqrt{2},-\sqrt{3}$$

Find the domain and the vertical and horizontal asymptotes (if any). $$f(x)=\frac{2}{x^{2}-9}$$

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$-x^{3}+x ; x-5$$

Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touch the \(x\) -axis? You may check your results with a graphing utility. $$g(x)=x^{3}(x+2)(x-3)$$

Complete them to review topics relevant to the remaining exercises. The graph of \(f(x)=(x-4)^{2}\) is the graph of \(y=x^{2}\) shifted __________ 4 units.

Solve the polynomial inequality. $$x^{3}-4 x^{2} \geq 0$$

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=x^{3}-5 x^{2}+8 x-4 ; x=2$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=2 x^{2}-5 x+3$$

Find the domain and the vertical and horizontal asymptotes (if any). $$f(x)=\frac{-x^{2}+9}{-2 x^{2}+8}$$

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$x^{3}+2 x^{2}-5 ; x^{2}-2$$

Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touch the \(x\) -axis? You may check your results with a graphing utility. $$f(x)=x^{2}+2 x+1$$

Solve the polynomial inequality. $$x^{3}+2 x^{2}+x<0$$

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=x^{3}-7 x+6 ; x=2$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=2 x^{2}-x-6$$

Find the domain and the vertical and horizontal asymptotes (if any). $$h(x)=\frac{-3 x^{2}+12}{x^{2}-9}$$

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$-x^{3}-3 x^{2}+6 ; x^{2}+1$$

Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touch the \(x\) -axis? You may check your results with a graphing utility. $$h(s)=s^{2}-2 s+1$$

Solve the polynomial inequality. $$x^{3}+5 x^{2}+4 x<0$$

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=-x^{4}-x^{3}+18 x^{2}+16 x-32 ; x=1$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{3}+5 x$$

Find the domain and the vertical and horizontal asymptotes (if any). $$h(x)=\frac{1}{(x-2)^{2}}$$

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$x^{5}-x^{4}+2 x^{3}+x^{2}-x+1 ; x^{3}+x-1$$

Solve the polynomial inequality. $$x^{3}+4 x^{2}+4 x<0$$

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=2 x^{3}-11 x^{2}+17 x-6 ; x=\frac{1}{2}$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{3}+7 x$$

Find the quotient and remainder when the first polynomial is divided by the second. You may use synthetic division wherever applicable. $$-2 x^{5}+x^{4}-x^{3}+2 x^{2}-1 ; x^{3}+x^{2}+1$$

Find the domain and the vertical and horizontal asymptotes (if any). $$G(x)=\frac{-2}{(x+4)^{2}}$$

Determine the multiplicities of the real zeros of the function. Comment on the behavior of the graph at the \(x\) -intercepts. Does the graph cross or just touch the \(x\) -axis? You may check your results with a graphing utility. $$h(x)=2 x^{3}-4 x^{2}+2 x$$

Solve the polynomial inequality. $$(x-2)\left(x^{2}-4\right)<0$$

Show that the given value of \(x\) is a zero of the polynomial. Use the zero to completely factor the polynomial. $$p(x)=3 x^{3}-2 x^{2}+3 x-2 ; x=\frac{2}{3}$$

Find all the zeros, real and nonreal, of the polynomial. Then express \(p(x)\) as a product of linear factors. $$p(x)=x^{2}-\pi^{2}$$

Write each polynomial in the form \(p(x)=d(x) q(x)+r(x),\) where \(p(x)\) is the given polynomial and \(d(x)\) is the given factor. You may use synthetic division wherever applicable. $$x^{2}+x+1 ; x+1$$

Find the domain and the vertical and horizontal asymptotes (if any). $$h(x)=\frac{3 x^{2}}{x+1}$$

Determine whether the function is a polynomial function. If so, find the degree. If not, state the reason. $$f(x)=-x^{3}+3 x^{3}+1$$

Determine what type of symmetry, if any, the function illustrates. Classify the function as odd, even, or neither. $$g(x)=x^{4}+2 x^{2}-1$$

Solve the polynomial inequality. $$(x-3)\left(x^{2}-25\right)<0$$