Chapter 3

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to verify that the expressions are equivalent. Verify the results algebraically. $$y_{1}=\frac{x^{4}-3 x^{2}-1}{x^{2}+5}, \quad y_{2}=x^{2}-8+\frac{39}{x^{2}+5}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$g(x)=x^{4}-4 x^{3}+8 x^{2}-16 x+16$$

Describe the graph of the quadratic function. Identify the vertex and \(x\) -intercept(s). Use a graphing utility to verify your results. \(f(x)=-2 x^{2}+16 x-31\)

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(h(t)=-\frac{2}{3}\left(t^{2}-5 t+3\right)\)

Determine the value that the function \(f\) approaches as the magnitude of \(x\) increases. Is \(f(x)\) greater than or less than this value when \(x\) is positive and large in magnitude? What about when \(x\) is negative and large in magnitude? $$f(x)=\frac{2 x-1}{x^{2}+1}$$

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$-2+4 i$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$h(x)=\frac{2 x-1}{x+5}$$

Use a graphing utility to graph the two equations in the same viewing window. Use the graphs to verify that the expressions are equivalent. Verify the results algebraically. $$y_{1}=\frac{x^{4}+x^{2}-1}{x^{2}+1}, \quad y_{2}=x^{2}-\frac{1}{x^{2}+1}$$

Find all the zeros of the function and write the polynomial as a product of linear factors. Use a graphing utility to verify your results graphically. (If possible, use the graphing utility to verify the imaginary zeros.) $$h(x)=x^{4}+6 x^{3}+10 x^{2}+6 x+9$$

Describe the graph of the quadratic function. Identify the vertex and \(x\) -intercept(s). Use a graphing utility to verify your results. \(f(x)=-4 x^{2}+24 x-41\)

Use the Leading Coefficient Test to describe the right-hand and left-hand behavior of the graph of the polynomial function. Use a graphing utility to verify your results. \(f(s)=-\frac{7}{8}\left(s^{3}+5 s^{2}-7 s+1\right)\)

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$f(x)=\frac{x^{2}-9}{x^{2}+5}$$

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$-5 i$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$g(x)=\frac{5}{x^{2}+1}$$

Write the function in the form \(f(x)=(x-k) q(x)+r(x)\) for the given value of \(k\). Use a graphing utility to demonstrate that \(f(k)=r\). Value of \(k\) \(k=4\) \(k=-\frac{2}{3}\) \(k=\sqrt{2}\) \(k=-\sqrt{5}\) \(k=1-\sqrt{3}\) \(k=2+\sqrt{2}\) Function $$f(x)=x^{3}-x^{2}-14 x+11$$

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=x^{2}-14 x+46$$

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(f(x)=3 x^{2}-12 x+3\)

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$h(x)=\frac{x^{3}+8}{x^{2}-11}$$

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$8 i$$

Write the function in the form \(f(x)=(x-k) q(x)+r(x)\) for the given value of \(k\). Use a graphing utility to demonstrate that \(f(k)=r\). Value of \(k\) \(k=4\) \(k=-\frac{2}{3}\) \(k=\sqrt{2}\) \(k=-\sqrt{5}\) \(k=1-\sqrt{3}\) \(k=2+\sqrt{2}\) Function $$f(x)=15 x^{4}+10 x^{3}-6 x^{2}+14$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$g(x)=-\frac{x}{(x-2)^{2}}$$

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=x^{2}-12 x+34$$

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(g(x)=5 x^{2}-10 x-5\)

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$g(x)=1+\frac{6}{x-3}$$

Write the function in the form \(f(x)=(x-k) q(x)+r(x)\) for the given value of \(k\). Use a graphing utility to demonstrate that \(f(k)=r\). Value of \(k\) \(k=4\) \(k=-\frac{2}{3}\) \(k=\sqrt{2}\) \(k=-\sqrt{5}\) \(k=1-\sqrt{3}\) \(k=2+\sqrt{2}\) Function $$f(x)=x^{3}+3 x^{2}-2 x-14$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$f(x)=\frac{x+1}{x^{2}-x-6}$$

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=2 x^{3}-3 x^{2}+8 x-12$$

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: (-2,5) Point: (0,9)

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(g(t)=\frac{1}{2} t^{4}-\frac{1}{2}\)

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$f(x)=3-\frac{-12}{x^{2}+2}$$

Write the function in the form \(f(x)=(x-k) q(x)+r(x)\) for the given value of \(k\). Use a graphing utility to demonstrate that \(f(k)=r\). Value of \(k\) \(k=4\) \(k=-\frac{2}{3}\) \(k=\sqrt{2}\) \(k=-\sqrt{5}\) \(k=1-\sqrt{3}\) \(k=2+\sqrt{2}\) Function $$f(x)=x^{3}+2 x^{2}-5 x-4$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$f(x)=\frac{x+4}{x^{2}+x-6}$$

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=2 x^{3}-5 x^{2}+18 x-45$$

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: (4,1) Point: (6,-7)

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(y=\frac{1}{4} x^{3}\left(x^{2}-9\right)\)

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$h(x)=\frac{x^{2}-x-20}{x^{2}+7}$$

Write the function in the form \(f(x)=(x-k) q(x)+r(x)\) for the given value of \(k\). Use a graphing utility to demonstrate that \(f(k)=r\). Value of \(k\) \(k=4\) \(k=-\frac{2}{3}\) \(k=\sqrt{2}\) \(k=-\sqrt{5}\) \(k=1-\sqrt{3}\) \(k=2+\sqrt{2}\) Function $$f(x)=4 x^{3}-6 x^{2}-12 x-4$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$f(x)=\frac{20 x}{x^{2}+1}-\frac{1}{x}$$

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=x^{3}-11 x+150$$

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: (1,-2)\(; \quad\) Point: (-1,14)

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(f(x)=x^{5}+x^{3}-6 x\)

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$g(x)=\frac{x^{2}-8 x+12}{x^{2}+4}$$

Write the function in the form \(f(x)=(x-k) q(x)+r(x)\) for the given value of \(k\). Use a graphing utility to demonstrate that \(f(k)=r\). Value of \(k\) \(k=4\) \(k=-\frac{2}{3}\) \(k=\sqrt{2}\) \(k=-\sqrt{5}\) \(k=1-\sqrt{3}\) \(k=2+\sqrt{2}\) Function $$f(x)=-3 x^{3}+8 x^{2}+10 x-8$$

Use a graphing utility to graph the function. Determine its domain and identify any vertical or horizontal asymptotes. $$f(x)=5\left(\frac{1}{x-4}-\frac{1}{x+2}\right)$$

(a) find all zeros of the function, (b) write the polynomial as a product of linear factors, and (c) use your factorization to determine the \(x\) -intercepts of the graph of the function. Use a graphing utility to verify that the real zeros are the only \(x\) -intercepts. $$f(x)=x^{3}+10 x^{2}+33 x+34$$

Write the standard form of the quadratic function that has the indicated vertex and whose graph passes through the given point. Use a graphing utility to verify your result. Vertex: (-4,-1) Point: (-2,4)

(a) find the zeros algebraically, (b) use a graphing utility to graph the function, and (c) use the graph to approximate any zeros and compare them with those from part (a). \(g(t)=t^{5}-6 t^{3}+9 t\)

Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. $$h(x)=\frac{2 x^{2}+11 x+5}{3 x^{2}+13 x-10}$$

Use a graphing utility to graph the function. What do you observe about its asymptotes? $$h(x)=\frac{6 x}{\sqrt{x^{2}+1}}$$

Use the Remainder Theorem and synthetic division to evaluate the function at each given value. Use a graphing utility to verify your results. \(f(x)=2 x^{3}-7 x+3\) (a) \(f(1)\) (b) \(f(-2)\) (c) \(f\left(\frac{1}{2}\right)\) (d) \(f(2)\)