Chapter 3

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}-2 x-4=0$$

Use synthetic division to find \(P(k)\). $$k=2 ; \quad P(x)=x^{2}-5 x+1$$

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$y=1.34 x^{2}-3 x+\sqrt{5}$$

Solve each problem. Path of a Frog's Leap Refer to Exercise \(37 .\) Suppose that the initial position of the frog is \((0,4)\) and its landing position is (6, 0). The height of the frog in feet is given by $$h(x)=-\frac{1}{3} x^{2}+\frac{4}{3} x+4$$ (a) What was the horizontal distance \(x\) from the base of the stump when the frog reached maximum height? (b) What was the maximum height?

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-70}}{\sqrt{-7}}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}+8 x+13=0$$

Without graphing, answer true or false to each statement. Then, support your answer by graphing. The function \(f(x)=x^{3}+3 x^{2}+3 x+1\) must have at least one real zero.

Use synthetic division to find \(P(k)\). $$k=3 ; \quad P(x)=x^{2}-x+3$$

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$y=-0.55 x^{2}+3.21 x \quad$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=2 x^{5}-10 x^{4}+x^{3}-5 x^{2}-x+5 \\\&=(x-5)\left(x^{2}+1\right)\left(2 x^{2}-1\right)\end{aligned}$$

Solve each problem. To determine the appropriate landing speed of Vangie's airplane, we might use $$f(x)=\frac{1}{10} x^{2}-3 x+22$$ where \(x\) is the initial landing speed in feet per second and \(f(x)\) is the length of the runway in feet. If the landing speed is too fast, she may run out of runway; if the speed is too slow, the plane may stall. If the runway is 800 feet long, what is the appropriate landing speed? What is the landing speed in mph? (Hint: 5280 feet = 1 mile)

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-24}}{\sqrt{8}}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}+2 x=-1$$

Without graphing, answer true or false to each statement. Then, support your answer by graphing. If a polynomial function of even degree has a negative leading coefficient and a positive \(y\) -value for its \(y\) -intercept, it must have at least two real zeros.

Use synthetic division to find \(P(k)\). $$k=0.5 ; \quad P(x)=x^{3}-x+4$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=3 x^{4}-4 x^{3}-22 x^{2}+15 x+18 \\\&=(3 x+2)(x-3)\left(x^{2}+x-3\right)\end{aligned}$$

Solve each problem. As a function of age group \(x,\) the fatality rate (per \(100,000\) population) for males killed in automobile accidents can be approximated by $$f(x)=1.8 x^{2}-12 x+37.4$$ where \(x=0\) represents ages \(21-24, x=1\) represents ages \(25-34, x=2\) represents ages \(35-44,\) and so on. Find the age group at which the accident rate is a minimum, and find the minimum rate. (Source: National Highway Traffic Safety Administration.)

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-54}}{\sqrt{27}}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$9 x^{2}-12 x=-8$$

Without graphing, answer true or false to each statement. Then, support your answer by graphing. The function \(f(x)=3 x^{4}+5\) has no real zeros.

Use synthetic division to find \(P(k)\). $$k=1.5 ; \quad P(x)=x^{3}+x-3$$

Graph each function in a viewing window that will allow you to use your calculator to approximate (a) the coordinates of the vertex and (b) the \(x\) -intercepts. Give values to the nearest hundredth. $$y=2.96 x^{2}+1.31$$

Solve each problem. The table lists the average heating bill for a natural gas consumer in Illinois during various months of the year. $$\begin{array}{c|c}\text { Month } & \text { Bill (\$) } \\\\\text { Jan. } & 108 \\\\\text { Mar. } & 68 \\\\\text { May } & 18 \\\\\text { July } & 12 \\\\\text { Sept. } & 13 \\\\\text { Nov. } & 54\end{array}$$ (a) Plot the data. Let \(x=1\) correspond to January, \(x=2\) to February, and so on. (b) Find a quadratic function \(f(x)=a(x-h)^{2}+k\) that models the data. Use \((7,12)\) as the vertex and \((1,108)\) as another point to determine \(a\) (c) Plot the data together with the graph of \(f\) in the same window. How well does \(f\) model the average heating bill over these months? (d) Use the quadratic regression feature of a graphing calculator to determine the quadratic function \(g\) that provides the best fit for the data. (e) Use the functions \(f\) and \(g\) to approximate the heating bill to the nearest dollar in the following months. (i) February (ii) June

Use synthetic division to find \(P(k)\) $$k=\sqrt{2} ; \quad P(x)=x^{4}-x^{2}-3$$

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-10}}{\sqrt{-40}}$$

Without graphing, answer true or false to each statement. Then, support your answer by graphing. The function \(f(x)=-3 x^{4}+5\) has two real zeros.

Solve each equation. For equations with real solutions, support your answers graphically. $$x(x-1)=1$$

Solve each problem. The table lists the projected number of shipments \(S\) of e-book readers in millions, \(x\) years after 2011 . $$\begin{array}{c|c}\text { Year } & S \\\0 & 23 \\\1 & 15 \\\2 & 11 \\\3 & 8 \\\4 & 7\end{array}$$ (a) Evaluate \(S(3)\) and interpret the result. (b) Find a quadratic function \(f\) to model these data. (c) Use \(f\) to estimate the number of shipments in 2017 to the nearest million. Do you think this is an accurate model for years past \(2015 ?\)

Use synthetic division to find \(P(k)\) $$k=\sqrt{3} ; \quad P(x)=x^{4}+2 x^{2}-10$$

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-40}}{\sqrt{20}}$$

Without graphing, answer true or false to each statement. Then, support your answer by graphing. The graph of \(f(x)=x^{3}-3 x^{2}+3 x-1=(x-1)^{3}\) has exactly one \(x\) -intercept.

Solve each equation. For equations with real solutions, support your answers graphically. $$x(x-3)=2$$

Solve each problem. Selected values of the stopping distance \(y\) in feet of a car traveling \(x\) mph are given in the table. $$\begin{array}{c|c}\begin{array}{c}\text { Speed } \\\\\text { (in mph) }\end{array} & \begin{array}{c} \text { Stopping Distance } \\\\\text { (in feet) }\end{array} \\\\\hline 20 & 46 \\\30 & 87 \\\40 & 140 \\\50 & 240 \\ 60 & 282 \\\70 & 371\end{array}$$ (a) Plot the data. (b) The quadratic function $$f(x)=0.056057 x^{2}+1.06657 x$$ is one model of the data. Find and interpret \(f(45)\) (c) Use a graph of the function in the same window as the data to determine how well \(f\) models the stopping distance.

Use synthetic division to find \(P(k)\) $$k=\sqrt[3]{4} ; \quad P(x)=-x^{3}+x+4$$

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-6} \cdot \sqrt{-2}}{\sqrt{3}}$$

Without graphing, answer true or false to each statement. Then, support your answer by graphing. A fifth-degree polynomial function cannot have a single real zero.

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}-5 x=x-7$$

Solve each equation and inequality. (a) \(3\left(x^{2}+4\right)+2 x(3 x-12)=0\) (b) \(3\left(x^{2}+4\right)+2 x(3 x-12)<0\)

Multiply or divide as indicated. Simplify each answer. $$\frac{\sqrt{-12} \cdot \sqrt{-6}}{\sqrt{8}}$$

Solve each problem. The coast-down time \(y\) for a typical car as it drops \(10 \mathrm{mph}\) from an initial speed \(x\) depends on several factors, such as average drag, tire pressure, and whether the transmission is in neutral. The table gives the coast-down time in seconds for a car under standard conditions for selected speeds in miles per hour. $$\begin{array}{c|c}\begin{array}{c}\text { Initial Speed } \\\\\text { (in mph) }\end{array} & \begin{array}{c}\text { Coast-Down Time } \\\\\text { (in seconds) }\end{array} \\\\\hline 30 & 30 \\\35 & 27 \\\40 & 23 \\\45 & 21 \\\50 &18 \\\55 & 16 \\\60 & 15 \\\65 & 13\end{array}$$ (a) Plot the data. (b) Use the quadratic regression feature of a graphing calculator to find the quadratic function \(g\) that best fits the data. Graph this function in the same window as the data. Is \(g\) a good model for the data? (c) Use \(g\) to predict the coast-down time at an initial speed of 70 mph. (d) Use the graph to find the speed that corresponds to a coast-down time of 24 seconds.

Without graphing, answer true or false to each statement. Then, support your answer by graphing. An even-degree polynomial function must have at least one real zero.

Solve each equation. For equations with real solutions, support your answers graphically. $$11 x^{2}-3 x+2=4 x+1$$

Solve each equation and inequality. (a) \(\left(x^{2}+3 x-1\right)+(2 x+3)(x-5)=0\) (b) \(\left(x^{2}+3 x-1\right)+(2 x+3)(x-5) \geq 0\)

Use synthetic division to determine whether the given number is a zero of the polynomial. $$2 ; \quad P(x)=x^{2}+2 x-8$$

Add or subtract as indicated. Write each sum or difference in standard form. $$(3+2 i)+(4-3 i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$4 x^{2}-12 x=-11$$

Solve each equation and inequality. (a) \(3(x+1)^{2}(2 x-1)^{4}+8(x+1)^{3}(2 x-1)^{3}=0\) (b) \(3(x+1)^{2}(2 x-1)^{4}+8(x+1)^{3}(2 x-1)^{3} \geq 0\)

Use synthetic division to determine whether the given number is a zero of the polynomial. $$-1 ; \quad P(x)=x^{2}+4 x-5$$

Add or subtract as indicated. Write each sum or difference in standard form. $$(4-i)+(2+5 i)$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=2 x-5$$