Chapter 3

Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$5 (\text { multiplicity } 2 )\text { and } -2 i$$

Write each number in simplest form, without a negative radicand. $$-11-\sqrt{-24}$$

Solve each equation analytically for all complex solutions, giving exact forms in your solution set. Then, graph the left side of the equation as \(y_{1}\) in the suggested viewing window and, using the capabilities of your calculator, support the real solutions. $$\begin{aligned} &-4 x^{3}-x^{2}+4 x=0\\\ &[-4,4] \text { by }[-10,10] \end{aligned}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x(12 x+11)=-2$$

It is not apparent from the standard viewing window whether the graph of the quadratic function intersects the \(x\) -axis once, twice, or not at all. Experiment with various windows to find the number of \(x\) -intercepts. If there are \(x\) -intercepts, give their coordinates to the nearest hundredth. (GRAPH CANT COPY) $$y=x^{2}+6.95 x+12.07$$

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=-3 x^{3}-x-5 ; \quad x+1$$

For each quadratic function defined , (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=2 x^{2}-8 x+9$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=2 x^{3}-5 x^{2}-x+6 \\\&=(x+1)(2 x-3)(x-2)\end{aligned}$$

Write each number in simplest form, without a negative radicand. $$i \sqrt{-9}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$-4+9 x-2 x^{2}=0$$

Solve each equation analytically for all complex solutions, giving exact forms in your solution set. Then, graph the left side of the equation as \(y_{1}\) in the suggested viewing window and, using the capabilities of your calculator, support the real solutions. $$\begin{aligned} &3 x^{3}+3 x^{2}+3 x=0\\\ &[-5,5] \text { by }[-5,5] \end{aligned}$$

It is not apparent from the standard viewing window whether the graph of the quadratic function intersects the \(x\) -axis once, twice, or not at all. Experiment with various windows to find the number of \(x\) -intercepts. If there are \(x\) -intercepts, give their coordinates to the nearest hundredth. (GRAPH CANT COPY) $$y=-x^{2}+6.5 x-10.60$$

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{5}-1 ; \quad x-1$$

For each quadratic function defined , (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=-3 x^{2}+24 x-46$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=x^{3}+x^{2}-8 x-12 \\\&=(x+2)^{2}(x-3)\end{aligned}$$

Solve each problem. Maximizing Revenue Suppose the revenue \(R\) in thousands of dollars that a company receives from producing \(x\) thousand MP3 players is \(R(x)=x(40-2 x)\) (a) Evaluate \(R(2)\) and interpret the result. (b) How many MP3 players should the company produce to maximize its revenue? (c) What is the maximum revenue?

Write each number in simplest form, without a negative radicand. $$i \sqrt{-16}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$-5+16 x-3 x^{2}=0$$

Solve each equation analytically for all complex solutions, giving exact forms in your solution set. Then, graph the left side of the equation as \(y_{1}\) in the suggested viewing window and, using the capabilities of your calculator, support the real solutions. $$\begin{aligned} &2 x^{3}+2 x^{2}+12 x=0\\\ &[-10,10] \text { by }[-20,20] \end{aligned}$$

It is not apparent from the standard viewing window whether the graph of the quadratic function intersects the \(x\) -axis once, twice, or not at all. Experiment with various windows to find the number of \(x\) -intercepts. If there are \(x\) -intercepts, give their coordinates to the nearest hundredth. (GRAPH CANT COPY) $$y=-2 x^{2}+0.2 x-0.15$$

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{7}+1 ; \quad x+1$$

For each quadratic function defined , (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=-2 x^{2}-6 x-5$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=x^{4}-18 x^{2}+81 \\\&=(x-3)^{2}(x+3)^{2}\end{aligned}$$

Solve each problem. The manager of an 80-unit apartment complex knows from experience that at a rent of \(\$ 400\) per month, all units will be rented. However, for each increase of \(\$ 20\) in rent, he can expect one unit to be vacated. Let \(x\) represent the number of \(\$ 20\) increases over \(\$ 400\). (a) Express, in terms of \(x,\) the number of apartments that will be rented if \(x\) increases of \(\$ 20\) are made. (For example, with three such increases, the number of apartments rented will be \(80-3=77\).) (b) Express the rent per apartment if \(x\) increases of \(\$ 20\) are made. (For example, if he increases rent by \(\$ 60=3 \times \$ 20,\) the rent per apartment is given by \(400+3(20)=\$ 460 .)\) (c) Determine a revenue function \(R\) in terms of \(x\) that will give the revenue generated as a function of the number of \(\$ 20\) increases. (d) For what number of increases will the revenue be \(\$ 37,500 ?\) (e) What rent should he charge in order to achieve the maximum revenue?

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-13} \cdot \sqrt{-13}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$\frac{1}{3} x^{2}-\frac{1}{3} x=24$$

Solve each equation analytically for all complex solutions, giving exact forms in your solution set. Then, graph the left side of the equation as \(y_{1}\) in the suggested viewing window and, using the capabilities of your calculator, support the real solutions. $$\begin{aligned} &x^{4}+17 x^{2}+16=0\\\ &[-4,4] \text { by }[-10,40] \end{aligned}$$

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

For each quadratic function defined , (a) use the vertex formula to find the coordinates of the vertex and (b) graph the function. Do not use a calculator. $$P(x)=2 x^{2}-2 x+1$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=x^{4}-8 x^{2}+16 \\\&=(x+2)^{2}(x-2)^{2}\end{aligned}$$

Solve each problem. When Respect Brings Success charges \(\$ 600\) for a seminar on management techniques, it attracts 1000 people. For each decrease of \(\$ 20\) in the charge, an additional 100 people will attend the seminar. Let \(x\) represent the number of \(\$ 20\) decreases in the charge. (a) Determine a revenue function \(R\) that will give revenue generated as a function of the number of \(\$ 20\) decreases. (b) Find the value of \(x\) that maximizes the revenue. What should the company charge to maximize the revenue? (c) What is the maximum revenue the company can generate?

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-17} \cdot \sqrt{-17}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$\frac{1}{6} x^{2}+\frac{1}{6} x=5$$

Solve each equation analytically for all complex solutions, giving exact forms in your solution set. Then, graph the left side of the equation as \(y_{1}\) in the suggested viewing window and, using the capabilities of your calculator, support the real solutions. $$\begin{aligned} &36 x^{4}+85 x^{2}+9=0\\\ &[-4,4] \text { by }[-10,40] \end{aligned}$$

Use synthetic division to find \(P(k)\). $$k=-2 ; \quad P(x)=x^{2}+5 x+6$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=2 x^{4}+x^{3}-6 x^{2}-7 x-2 \\\&=(2 x+1)(x-2)(x+1)^{2}\end{aligned}$$

Solve each problem. Shooting a Foul Shot To make a foul shot in basketball, the ball must follow a parabolic arc that depends on both the angle and velocity with which the basketball is released. If a person shoots the basketball overhand from a position 8 feet above the floor, then the path can sometimes be modeled by the quadratic function $$f(x)=\frac{-16 x^{2}}{0.434 v^{2}}+1.15 x+8$$ where \(v\) is the initial velocity of the ball in feet per second, as illustrated in the figure. (Source: Rist, C., "The Physics of Foul Shots," Discover, October 2000.) CAN'T COPY THE IMAGE (a) If the basketball hoop is 10 feet high and located 15 feet away, what initial velocity \(v\) should the basketball have? (b) Check your answer from part (a) graphically. Plot the point \((0,8)\) where the ball is released and the point \((15,10)\) where the basketball hoop is. Does your graph pass through both points? (c) What is the maximum height of the basketball?

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-3} \cdot \sqrt{-8}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+2)(x-1)=7 x+5$$

Solve each equation analytically for all complex solutions, giving exact forms in your solution set. Then, graph the left side of the equation as \(y_{1}\) in the suggested viewing window and, using the capabilities of your calculator, support the real solutions. $$\begin{aligned} &x^{6}+19 x^{3}-216=0\\\ &[-4,4] \text { by }[-350,200] \end{aligned}$$

Use synthetic division to find \(P(k)\). $$k=-2 ; \quad P(x)=5 x^{3}+2 x^{2}-x+5$$

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. $$P(x)=-0.32 x^{2}+\sqrt{3} x+2.86$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=3 x^{4}-7 x^{3}-6 x^{2}+12 x+8 \\\&=(3 x+2)(x+1)(x-2)^{2}\end{aligned}$$

Multiply or divide as indicated. Simplify each answer. $$\sqrt{-5} \cdot \sqrt{-15}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(x+4)(x-1)=-5 x-4$$

Solve each equation analytically for all complex solutions, giving exact forms in your solution set. Then, graph the left side of the equation as \(y_{1}\) in the suggested viewing window and, using the capabilities of your calculator, support the real solutions. $$\begin{aligned} &8 x^{6}+7 x^{3}-1=0\\\ &[-4,4] \text { by }[-5,100] \end{aligned}$$

Use synthetic division to find \(P(k)\). $$k=2 ; \quad P(x)=2 x^{3}-3 x^{2}-5 x+4$$

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. $$P(x)=-\sqrt{2} x^{2}+0.45 x+1.39$$

Draw by hand a rough sketch of the graph of each function. (You may wish to support your answer with a calculator graph.) $$\begin{aligned}P(x) &=x^{4}+3 x^{3}-3 x^{2}-11 x-6 \\\&=(x+3)(x+1)^{2}(x-2)\end{aligned}$$

Solve each problem. Path of a Frog's Leap A frog leaps from a stump 3 feet high and lands 4 feet from the base of the stump. We can consider the initial position of the frog to be at \((0,3)\) and its landing position to be at \((4,0)\) CAN'T COPY THE IMAGE It is determined that the height \(h\) in feet of the frog as a function of its distance \(x\) from the base of the stump is given by $$h(x)=-0.5 x^{2}+1.25 x+3$$ (a) How high was the frog when its horizontal distance \(x\) from the base of the stump was 2 feet? (b) What was the horizontal distance from the base of the stump when the frog was 3.25 feet above the ground? (c) At what horizontal distance from the base of the stump did the frog reach its highest point? (d) What was the maximum height reached by the frog?