Chapter 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.) $$(-4,0),(0,0)$$

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

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$0,0,4,1+i$$

Perform the indicated operation and write the result in standard form. $$(\sqrt{14}+\sqrt{10} i)(\sqrt{14}-\sqrt{10} i)$$

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

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=2 x^{3}+x^{2}-14 x-10, \quad k=1+\sqrt{3}$$

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. $$f(x)=2 x^{4}-2 x^{2}-40$$

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.) $$(0,0),(10,0)$$

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$C(x)=\frac{5+2 x}{1+x}$$

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$\frac{2}{3},-1,3+\sqrt{2} i$$

Perform the indicated operation and write the result in standard form. $$(2-\sqrt{-8})(8+\sqrt{-6})$$

Use the zero or root feature of a graphing utility to approximate the real zeros of \(f\). Give your approximations to the nearest thousandth. $$f(x)=7 x^{4}-42 x^{3}+43 x^{2}+216 x-324$$

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. $$g(t)=t^{5}-6 t^{3}+9 t$$

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.) $$(4,0),(8,0)$$

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

Find a polynomial with real coefficients that has the given zeros. (There are many correct answers.) $$\frac{3}{4},-2,-\frac{1}{2}+i$$

Perform the indicated operation and write the result in standard form. $$(3+\sqrt{-5})(7-\sqrt{-10})$$

Use the zero or root feature of a graphing utility to approximate the real zeros of \(f\). Give your approximations to the nearest thousandth. $$f(x)=3 x^{4}-12 x^{3}+27 x^{2}+4 x-4$$

Write the function in the form \(f(x)=(x-k) q(x)+r\) for the given value of \(k\), and demonstrate that \(f(k)=r\). $$f(x)=3 x^{3}-19 x^{2}+27 x-7, \quad k=3-\sqrt{2}$$

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. $$f(x)=x^{3}-3 x^{2}+2 x-6$$

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.) $$(-3,0),\left(-\frac{1}{2}, 0\right)$$

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

Write the quotient in standard form. $$\frac{3-i}{3+i}$$

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ; \quad\) Irrational zeros: 0 (c) Rational zeros: 1; Irrational zeros: 2 (d) Rational zeros: 1; Irrational zeros: 0 $$f(x)=x^{3}-1$$

Use synthetic division to find each function value. \(f(x)=2 x^{5}-3 x^{2}-4 x-1\) (a) \(f(-2)\) (b) \(f(-4)\) (c) \(f(1)\) (d) \(f(3)\)

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. $$f(x)=x^{3}-4 x^{2}-25 x+100$$

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.) $$\left(-\frac{5}{2}, 0\right),(2,0)$$

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

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$x^{4}-6 x^{2}-72$$

Write the quotient in standard form. $$\frac{5}{4-2 i}$$

Use synthetic division to find each function value. \(g(x)=x^{6}-4 x^{4}+3 x^{2}+2\) (a) \(g(2)\) (b) \(g(-4)\) (c) \(g(7)\) (d) \(g(-1)\)

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(x)=\frac{2}{3} x+5$$

Optimal Area The perimeter of a rectangle is 200 feet. Let \(x\) represent the width of the rectangle. Write a quadratic function for the area of the rectangle in terms of its width. Find the vertex of the graph of the quadratic function and interpret its meaning in the context of the problem.

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

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$x^{4}-5 x^{3}+4 x^{2}+x-15$$

Write the quotient in standard form. $$\frac{5}{4-2 i}$$

Match the cubic function with the numbers of rational and irrational zeros. (a) Rational zeros: \(0 ; \quad\) Irrational zeros: 1 (b) Rational zeros: \(3 ; \quad\) Irrational zeros: 0 (c) Rational zeros: 1; Irrational zeros: 2 (d) Rational zeros: 1; Irrational zeros: 0 $$f(x)=x^{3}-x$$

Use synthetic division to find each function value. \(f(x)=2 x^{3}-3 x^{2}+8 x-14\) (a) \(f(2)\) (b) \(f(-1)\) (c) \(f(1.1)\) (d) \(f(3)\)

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$h(x)=-\frac{3}{4} x+2$$

Optimal Area The perimeter of a rectangle is 540 feet. Let \(x\) represent the width of the rectangle. Write a quadratic function for the area of the rectangle in terms of its width. Find the vertex of the graph of the quadratic function and interpret its meaning in the context of the problem.

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

Write the polynomial (a) as the product of factors that are irreducible over the rationals, (b) as the product of linear and quadratic factors that are irreducible over the reals, and (c) in completely factored form. $$x^{4}+x^{3}+8 x^{2}+9 x-9$$

Write the quotient in standard form. $$\frac{3}{1+2 i}$$

Use synthetic division to find each function value. \(f(x)=3 x^{4}-7 x^{3}+5 x-12\) (a) \(f(1)\) (b) \(f(4)\) (c) \(f(-3)\) (d) \(f(-1.2)\)

Analyzing a Graph In Exercises \(47-58\), analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$f(t)=\frac{1}{2}\left(t^{2}-4 t-1\right)$$

Sketch the graph of the rational function. To aid in sketching the graphs, check for intercepts, symmetry, vertical asymptotes, and horizontal asymptotes. $$h(x)=\frac{x^{2}}{x^{2}-9}$$

Write the quotient in standard form. $$\frac{7+10 i}{i}$$

Dimensions of a Box An open box is to be made from a rectangular piece of material, 18 inches by 15 inches, by cutting equal squares from the corners and turning up the sides (see figure). (a) Write the volume \(V\) of the box as a function of \(x\). Determine the domain of the function. (b) Sketch the graph of the function and approximate the dimensions of the box that yield a maximum volume. (c) Find values of \(x\) such that \(V=108 .\) Which of these values is a physical impossibility in the construction of the box? Explain. (d) What value of \(x\) should you use to make the tallest possible box with a volume of 108 cubic inches?

Use synthetic division to find each function value. \(f(x)=1.2 x^{3}-0.5 x^{2}-2.1 x-2.4\) (a) \(f(2)\) (b) \(f(-6)\) (c) \(f\left(\frac{2}{3}\right)\) (d) \(f(1)\)

Analyze the graph of the function algebraically and use the results to sketch the graph by hand. Then use a graphing utility to confirm your sketch. $$g(x)=-x^{2}+10 x-16$$