Chapter 3

Use long division to divide and use the result to factor the dividend completely. $$\left(5 x^{2}-17 x-12\right) \div(x-4)$$

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=x^{2}-1\)

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model. (0,2795),(5,1590),(10,650),(15,-30),(20,-450) \((25,-615), \quad(30,-520), \quad(35,-55), \quad(40,625)\) (45,1630),(50,2845),(55,4350)

Use a graphing utility to graph \(f(x)=2 / x\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=-f(x)$$

Confirm that the function has the indicated zeros. $$f(x)=3 x^{4}-48 ;-2,2,-2 i, 2 i$$

Use long division to divide and use the result to factor the dividend completely. $$\left(x^{3}+5 x^{2}-12 x-36\right) \div(x+2)$$

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=(x+3)^{2}\)

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model. (0,6140),(2,6815),(4,7335),(6,7710),(8,7915) \((10,7590), \quad(12,7975), \quad(14,7700)\) (16,7325) (18,6820),(20,6125),(22,5325)

Use a graphing utility to graph \(f(x)=2 / x\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=\frac{1}{2} f(x+2)$$

Confirm that the function has the indicated zeros. $$f(x)=2 x^{5}-2 x ; 0,1,-1-i, i$$

Use long division to divide and use the result to factor the dividend completely. $$\left(2 x^{3}-3 x^{2}-50 x+75\right) \div(2 x-3)$$

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=-(x+3)^{2}-1\)

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model. $$\begin{aligned} &(1,4.0),(2,6.5),(3,8.8),(4,10.6),(5,13.9),(6,15.0)\\\ &(7,17.5),(8,20.1),(9,24.0),(10,27.1) \end{aligned}$$

Use a graphing utility to graph \(f(x)=3 / x^{2}\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=f(x)-2$$

Find all the zeros of the function. Is there a relationship between the number of real zeros and the number of \(x\) -intercepts of the graph? Explain. $$\begin{aligned} f(x)=& x^{3}-4 x^{2} \\ &+x-4 \end{aligned}$$

Use long division to divide. $$\left(x^{3}-4 x^{2}-17 x+6\right) \div(x-3)$$

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=(x+1)^{2}\)Graph

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model. $$\begin{aligned} &(-6,10.7), \quad(-4,9.0), \quad(-2,7.0), \quad(0,5.4), \quad(2,3.5)\\\ &(4,1.7),(6,-0.1),(8,-1.8),(10,-3.6),(12,-5.3) \end{aligned}$$

Use a graphing utility to graph \(f(x)=3 / x^{2}\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=-2 f(x)$$

Find all the zeros of the function. Is there a relationship between the number of real zeros and the number of \(x\) -intercepts of the graph? Explain. $$\begin{aligned} f(x)=& x^{3}-4 x^{2} \\ &-4 x+16 \end{aligned}$$

Use long division to divide. $$\left(4 x^{3}-7 x^{2}-11 x+5\right) \div(4 x+5)$$

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=-x^{2}+2\)

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model. \((0,587), \quad(5,551), \quad(10,512), \quad(15,478), \quad(20,436)\) (25,430),(30,424),(35,420),(40,423),(45,429) (50,444)

Use a graphing utility to graph \(f(x)=3 / x^{2}\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=f(x-2)$$

Find all the zeros of the function. Is there a relationship between the number of real zeros and the number of \(x\) -intercepts of the graph? Explain. $$f(x)=x^{4}+4 x^{2}+4$$

Use long division to divide. $$\left(7 x^{3}+3\right) \div(x+2)$$

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=(x-3)^{2}\)

(a) use a graphing utility to create a scatter plot of the data, (b) determine whether the data could be better modeled by a linear model or a quadratic model, (c) use the regression feature of the graphing utility to find a model for the data, (d) use the graphing utility to graph the model with the scatter plot from part (a), and (e) create a table comparing the original data with the data given by the model. $$\begin{aligned} &(2,34.3), \quad(3,33.8), \quad(4,32.6), \quad(5,30.1), \quad(6,27.8)\\\ &(9,14.8), \quad(10,9.4), \quad(11,3.7)\\\ &(7,22.5), \quad(8,19.1)\\\ &(12,-1.6) \end{aligned}$$

Use a graphing utility to graph \(f(x)=3 / x^{2}\) and the function \(g\) in the same viewing window. Describe the relationship between the two graphs. $$g(x)=\frac{1}{4} f(x)$$

Use long division to divide. $$\left(8 x^{4}-5\right) \div(2 x+1)$$

Find all the zeros of the function. Is there a relationship between the number of real zeros and the number of \(x\) -intercepts of the graph? Explain. $$f(x)=x^{4}-3 x^{2}-4$$

Sketch the graph of the function and compare it with the graph of \(y=x^{2}\) \(y=-(x-3)^{2}+1\)

Find any asymptotes of the graph of the rational function. Verify your answers by using a graphing utility to graph the function. $$f(x)=\frac{1}{x^{2}}$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{1}{x+2}$$

Use long division to divide. $$\left(5 x-1+10 x^{3}-2 x^{2}\right) \div\left(2 x^{2}+1\right)$$

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.) $$h(x)=x^{2}-4 x+1$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=20-x^{2}\)

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) \(g(x)=(x-3)^{3}\)

Find any asymptotes of the graph of the rational function. Verify your answers by using a graphing utility to graph the function. $$f(x)=\frac{3}{(x-2)^{3}}$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$f(x)=\frac{1}{x-6}$$

Use long division to divide. $$\left(1+3 x^{2}+x^{4}\right) \div\left(3-2 x+x^{2}\right)$$

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.) $$g(x)=x^{2}+10 x+23$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=x^{2}+8\)

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) \(g(x)=x^{3}-3\)

Find any asymptotes of the graph of the rational function. Verify your answers by using a graphing utility to graph the function. $$f(x)=\frac{2 x^{2}}{x^{2}+x-6}$$

Sketch the graph of the rational function by hand. As sketching aids, check for intercepts, vertical asymptotes, horizontal asymptotes, and holes. Use a graphing utility to verify your graph. $$C(x)=\frac{5+2 x}{1+x}$$

Use long division to divide. $$\left(x^{3}-9\right) \div\left(x^{2}+1\right)$$

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.) $$f(x)=x^{2}-12 x+26$$

Describe the graph of the function and identify the vertex. Use a graphing utility to verify your results. \(f(x)=\frac{1}{2} x^{2}-5\)

Sketch the graph of \(f(x)=x^{3}\) and the graph of the function \(g .\) Describe the transformation from \(f\) to \(g .\) \(g(x)=-x^{3}+4\)