Chapter 3

Use the Intermediate Value Theorem to show that the function has at least one zero in the interval \([a, b] .\) (You do not have to approximate the zero.) $$f(x)=-x^{3}+2 x^{2}+7 x-3, \quad[3,4]$$

Use synthetic division to divide. Divisor \(x-6\) Dividend $$10 x^{4}-50 x^{3}-800$$

Determine (a) the maximum number of turning points of the graph of the function and (b) the maximum number of real zeros of the function. $$f(x)=x^{2}-4 x+1$$

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$f(x)=-\left(x^{2}+2 x-3\right)$$

Compare the graph of \(f(x)=8 / x^{3}\) with the graph of \(g\). $$g(x)=f(x)+5=\frac{8}{x^{3}}+5$$

Find all the zeros of the function and write the polynomial as a product of linear factors. $$f(x)=5 x^{3}-9 x^{2}+28 x+6$$

Perform the indicated operation and write the result in standard form. $$(3+4 i)^{2}+(3-4 i)^{2}$$

Use the Intermediate Value Theorem to approximate the zero of \(f\) in the interval \([a, b]\). Give your approximation to the nearest tenth. (If you have a graphing utility, use it to help you approximate the zero.) $$f(x)=x^{3}+x-1, \quad[0,1]$$

Use synthetic division to divide. Divisor \(x+3\) Dividend $$x^{5}-13 x^{4}-120 x+80$$

Determine (a) the maximum number of turning points of the graph of the function and (b) the maximum number of real zeros of the function. $$f(x)=-3 x^{4}+1$$

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$f(x)=-\left(x^{2}+6 x-3\right)$$

Compare the graph of \(f(x)=8 / x^{3}\) with the graph of \(g\). $$g(x)=f(x-3)=\frac{8}{(x-3)^{3}}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. $$g(x)=3 x^{3}-4 x^{2}+8 x+8$$

Perform the indicated operation and write the result in standard form. $$(2-5 i)^{2}-(2+5 i)^{2}$$

Use the Intermediate Value Theorem to approximate the zero of \(f\) in the interval \([a, b]\). Give your approximation to the nearest tenth. (If you have a graphing utility, use it to help you approximate the zero.) $$f(x)=x^{5}+x+1, \quad[-1,0]$$

Use synthetic division to divide. Divisor \(x-4\) Dividend $$2 x^{5}-30 x^{3}-37 x+13$$

Determine (a) the maximum number of turning points of the graph of the function and (b) the maximum number of real zeros of the function. $$f(x)=-x^{5}+x^{4}-x$$

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$f(x)=x^{2}-x+\frac{5}{4}$$

Compare the graph of \(f(x)=8 / x^{3}\) with the graph of \(g\). $$g(x)=-f(x)=-\frac{8}{x^{3}}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. $$g(x)=x^{4}-4 x^{3}+8 x^{2}-16 x+16$$

Perform the indicated operation and write the result in standard form. $$\sqrt{-3} \cdot \sqrt{-8}$$

Use the Intermediate Value Theorem to approximate the zero of \(f\) in the interval \([a, b]\). Give your approximation to the nearest tenth. (If you have a graphing utility, use it to help you approximate the zero.) $$f(x)=x^{4}-10 x^{2}-11, \quad[3,4]$$

Use synthetic division to divide. Divisor \(x+3\) Dividend $$5 x^{3}$$

Determine (a) the maximum number of turning points of the graph of the function and (b) the maximum number of real zeros of the function. $$f(x)=2 x^{3}+x^{2}+1$$

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$f(x)=x^{2}+3 x+\frac{1}{4}$$

Compare the graph of \(f(x)=8 / x^{3}\) with the graph of \(g\). $$g(x)=\frac{1}{4} f(x)=\frac{2}{x^{3}}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. $$h(x)=x^{4}+6 x^{3}+10 x^{2}+6 x+9$$

Perform the indicated operation and write the result in standard form. $$\sqrt{-5} \cdot \sqrt{-10}$$

Use the Intermediate Value Theorem to approximate the zero of \(f\) in the interval \([a, b]\). Give your approximation to the nearest tenth. (If you have a graphing utility, use it to help you approximate the zero.) $$f(x)=-x^{3}+3 x^{2}+9 x-2, \quad[4,5]$$

Use synthetic division to divide. Divisor \(x-2\) Dividend $$-3 x^{4}$$

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$f(x)=-x^{2}+2 x+5$$

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{1}{x+3}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. $$f(x)=x^{4}+10 x^{2}+9$$

Perform the indicated operation and write the result in standard form. $$(\sqrt{-10})^{2}$$

Use synthetic division to divide. Divisor \(x+3\) Dividend $$2 x^{5}$$

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$f(x)=x^{2}-25$$

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$f(x)=-x^{2}-4 x+1$$

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{1}{x-3}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. $$f(x)=x^{4}+29 x^{2}+100$$

Perform the indicated operation and write the result in standard form. $$(\sqrt{-75})^{3}$$

Use synthetic division to divide. Divisor \(x+1\) Dividend $$5-3 x+2 x^{2}-x^{3}$$

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$h(t)=t^{2}+8 t+16$$

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$h(x)=4 x^{2}-4 x+21$$

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{1}{x-4}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. $$f(t)=t^{5}+5 t^{4}-7 t^{3}-43 t^{2}-8 t-48$$

Perform the indicated operation and write the result in standard form. $$(2+3 i)(1-i)$$

Use synthetic division to divide. Divisor \(x-6\) Dividend $$180 x-x^{4}$$

Algebraic and Graphical Approaches In Exercises \(31-46\), find all real zeros of the function algebraically. Then use a graphing utility to confirm your results. $$f(x)=x^{2}-12 x+36$$

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$f(x)=2 x^{2}-x+1$$

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{1}{x+6}$$