Chapter 3

Find all the zeros of the function and write the polynomial as a product of linear factors. $$g(x)=x^{5}-8 x^{4}+28 x^{3}-56 x^{2}+64 x-32$$

Perform the indicated operation and write the result in standard form. $$(6-5 i)(1+i)$$

Use synthetic division to divide. Divisor \(x+\frac{1}{2}\) Dividend $$4 x^{3}+16 x^{2}-23 x-15$$

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. $$f(x)=\frac{1}{3} x^{2}+\frac{1}{3} x-\frac{2}{3}$$

Sketch the graph of the quadratic function. Identify the vertex and intercepts. $$f(x)=\frac{1}{4}\left(x^{2}-16 x+32\right)$$

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

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$-2,3 i,-3 i$$

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

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

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. $$f(x)=\frac{1}{2} x^{2}+\frac{5}{2} x-\frac{3}{2}$$

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

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{-2}{x-3}$$

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$5,2 i,-2 i$$

Perform the indicated operation and write the result in standard form. $$(8+3 i)(8-3 i)$$

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=x^{3}+x^{2}-12 x+20, \quad k=3$$

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. $$f(x)=2 x^{2}+4 x+6$$

Find an equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex: \((2,-1) ;\) point: \((4,-3)\)

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{x+4}{x-5}$$

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$1,2+i, 2-i$$

Perform the indicated operation and write the result in standard form. $$5 i(4-6 i)$$

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

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=x^{3}-2 x^{2}-15 x+7, \quad k=-4$$

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. $$g(x)=-5\left(x^{2}+2 x-4\right)$$

Find an equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex: \((-3,5)\); point: \((-6,-1)\)

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{x-2}{x-3}$$

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$6,-5+2 i,-5-2 i$$

Perform the indicated operation and write the result in standard form. $$-2 i(7+9 i)$$

Use the zoom and trace features of a graphing utility to approximate the real zeros of \(f\). Give your approximations to the nearest thousandth. $$f(x)=4 x^{3}+14 x-8$$

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

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. $$f(t)=t^{3}-4 t^{2}+4 t$$

Find an equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex: \((5,12) ;\) point: \((7,15)\)

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{2+x}{1-x}$$

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$-4,3 i,-3 i, 2 i,-2 i$$

Perform the indicated operation and write the result in standard form. $$(5+6 i)^{2}$$

Use the zoom and trace features of a graphing utility to approximate the real zeros of \(f\). Give your approximations to the nearest thousandth. $$f(x)=x^{3}-3.9 x^{2}+4.79 x-1.881$$

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

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. $$f(x)=x^{4}-x^{3}-20 x^{2}$$

Find an equation of the parabola that has the indicated vertex and whose graph passes through the given point. Vertex: \((-2,-2) ;\) point: \((-1,0)\)

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$f(x)=\frac{3-x}{2-x}$$

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$2,2,2,4 i,-4 i$$

Perform the indicated operation and write the result in standard form. $$(3-7 i)^{2}$$

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 \(f(k)=r\). $$f(x)=x^{3}+2 x^{2}-3 x-12, \quad k=\sqrt{3}$$

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

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.) $$(2,0),(-1,0)$$

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

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

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

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

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. $$f(x)=\frac{1}{3}-\frac{1}{3} x^{2}$$