91Ó°ÊÓ

Use the zoom and trace features of a graphing utility to approximate the real zeros of \(f\). Give your approximations to the nearest thousandth. $$f(x)=-x^{3}+2 x^{2}+4 x+5$$

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=4 x^{4}+6 x^{3}+4 x^{2}-5 x+13, \quad k=-\frac{1}{2}$$

Perform the indicated operation and write the result in standard form. $$(\sqrt{5}-\sqrt{3} i)(\sqrt{5}+\sqrt{3} i)$$

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=x^{3}+2 x^{2}-3 x-12, \quad k=\sqrt{3}$$

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

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. $$g(t)=\frac{1}{2} t^{4}-\frac{1}{2}$$

Use the zero or root feature of a graphing utility to approximate the real zeros of \(f\). Give your approximations to the nearest thousandth. $$f(x)=x^{4}+x-3$$

Find two quadratic functions whose graphs have the given \(x\) -intercepts. Find one function whose graph opens upward and another whose graph opens downward. (There are many correct answers.) $$(2,0),(-1,0)$$

Use the zero or root feature of a graphing utility to approximate the real zeros of \(f\). Give your approximations to the nearest thousandth. $$f(x)=-x^{4}+2 x^{3}+4$$

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=x^{3}+3 x^{2}-7 x-6, \quad k=-\sqrt{2}$$