Chapter 3

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=-x^{4}+2 x^{3}+x+12 ; \quad 2.7 \text { and } 2.8$$

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=x^{5}-x^{4}-\pi x^{6}-x+3$$

Solve each problem. Area of a Parking Lot American River College has plans to construct a rectangular parking lot on land bordered on one side by a highway. There are 640 feet of fencing available to fence the other three sides. Let \(x\) represent the length of each of the two parallel sides of fencing. (a) Express the length of the remaining side to be fenced in terms of \(x\) (b) What are the restrictions on \(x ?\) (c) Determine a function \(\mathscr{A}\) ) that represents the area of the parking lot in terms of \(x .\) (d) Graph the function \(\mathscr{A}\) from part (c) in a viewing window of \([0,320]\) by \([0,55,000] .\) Determine graphically the values of \(x\) that will give an area between \(30,000\) and \(40,000\) square feet. (e) What dimensions will give a maximum area, and what will this area be? Determine analytically and support graphically. (IMAGE CAN'T COPY)

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=3 x^{2}+4 x-1$$

Determine whether each statement is true or false. If is false, tell why. No real number is a pure imaginary number.

Find all complex solutions of each equation. $$5 x^{3}-x^{2}+10 x-2=0$$

Solve each equation. For equations with real solutions, support your answers graphically. $$2 x^{2}=48$$

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=-2 x^{4}+x^{3}-x^{2}+3 ; \quad-1 \text { and }-0.9$$

Solve each problem. Area of a Rectangular Region A farmer wishes to enclose a rectangular region bordering a barn with fencing, as shown in the diagram. Suppose that \(x\) represents the length of each of the three parallel pieces of fencing. She has 600 feet of fencing available. (a) What is the length of the remaining piece of fencing in terms of \(x ?\) (b) Determine a function \(\mathscr{A}\) that represents the total area of the enclosed region. Give any restrictions on \(x\) (c) What dimensions for the total enclosed region would give an area of \(22,500\) square feet? Determine the answer analytically. (d) Use a graph to find the maximum area that can be enclosed. (IMAGE CAN'T COPY)

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=-x-3.2 x^{3}+x^{2}-2.84 x^{4}$$

For each quadratic function defined , (a) write the function in the form \(P(x)=a(x-h)^{2}+k,\) (b) give the vertex of the parabola, and (c) graph the function. Do not use a calculator. $$P(x)=4 x^{2}+3 x-1$$

Determine whether each statement is true or false. If is false, tell why. Every pure imaginary number is a complex number.

Find all complex solutions of each equation. $$x^{4}-11 x^{2}+10=0$$

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

Use the intermediate value theorem to show that each function has a real zero between the two numbers given. Then, use your calculator to approximate the zero to the nearest hundredth. $$P(x)=x^{5}-2 x^{3}+1 ;-1.6 \text { and }-1.5$$

Solve each problem. Hitting a Baseball A baseball is hit so that its height in feet after \(t\) seconds is $$ s(t)=-16 t^{2}+44 t+4 $$ (a) How high is the baseball after 1 second? (b) Find the maximum height of the baseball.

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=x^{10,000}$$

\(\quad\) Match each equation in Column I with the description of the parabola that is its graph in Column II. 1 (a) \(y=(x-4)^{2}-2 \quad\) A. Vertex \((2,-4),\) opens downward (b) \(y=(x-2)^{2}-4\) B. Vertex \((2,-4),\) opens upward (c) \(y=-(x-4)^{2}-2\) C. Vertex \((4,-2),\) opens downward (d) \(y=-(x-2)^{2}-4\) D. Vertex \((4,-2),\) opens upward

For each polynomial, one or more zeros are given. Find all remaining zeros. \(P(x)=2 x^{4}-2 x^{3}+55 x^{2}-50 x+125 ; \quad-5 i\) is a zero.

Determine whether each statement is true or false. If is false, tell why. A number can be both real and complex.

Find all complex solutions of each equation. $$x^{4}+x^{2}-6=0$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-100$$

Solve each problem. Hitting a Golf Ball A golf ball is hit so that its height \(h\) in feet after \(t\) seconds is $$ h(t)=-16 t^{2}+60 t $$ (a) What is the initial height of the golf ball? (b) How high is the golf ball after 1.5 seconds? (c) Find the maximum height of the golf ball.

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=-x^{104,266}$$

Concept Check Match each equation in Column I with the description of the parabola that is its graph in Column II, assuming \(a>0, h>0,\) and \(k>0\) (a) \(y=-a(x+h)^{2}+k\) A. Vertex in quadrant I, two \(x\) -intercepts (b) \(y=a(x-h)^{2}+k\) B. Vertex in quadrant I, no \(x\) -intercepts (c) \(y=a(x+h)^{2}+k\) C. Vertex in quadrant II, two \(x\) -intercepts (d) \(y=-a(x-h)^{2}+k\) D. Vertex in quadrant II, no \(x\) -intercepts

Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$5 \text { and } -4$$

Determine whether each statement is true or false. If is false, tell why. There is no real number that is a complex number.

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. $$\begin{aligned} &4 x^{4}-25 x^{2}+36=0\\\ &[-5,5] \text { by }[-5,100] \end{aligned}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-18$$

Suppose that a polynomial function \(P\) is defined in such a way that \(P(2)=-4\) and \(P(2.5)=2\) What conclusion does the intermediate value theorem allow you to make?

Solve each problem. Height of a Baseball A baseball is dropped from a stadium seat that is 75 feet above the ground. Its height \(s\) in feet after \(t\) seconds is given by $$ s(t)=75-16 t^{2} $$ Estimate to the nearest tenth of a second how long it takes for the baseball to strike the ground.

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=-3 x^{15,297}$$

Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$6 \text { and } -2$$

Determine whether each statement is true or false. If is false, tell why. A complex number might not be a pure imaginary number.

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. $$\begin{aligned} &4 x^{4}-29 x^{2}+25=0\\\ &[-5,5] \text { by }[-50,100] \end{aligned}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=-32$$

Suppose that a polynomial function \(P\) is defined in such a way that \(P(3)=-4\) and \(P(4)=-10 .\) Can we be certain that there is no zero between 3 and \(4 ?\) Explain, using a graph.

Solve each problem. Geometry \(\quad\) A cylindrical aluminum can is being constructed to have a height \(h\) of 4 inches. If the can is to have a volume of 28 cubic inches, approximate its radius \(r\) (Hint: \(\left.V=\pi r^{2} h .\right)\)

Use an end behavior diagram or to describe the end behavior of the graph of each function. Do not use a calculator. $$P(x)=12 x^{107,499}$$

Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$-3,2, \text{ and } i$$

Write each number in simplest form, without a negative radicand. $$\sqrt{-100}$$

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. $$\begin{aligned} &x^{4}-15 x^{2}-16=0\\\ &[-5,5] \text { by }[-100,100] \end{aligned}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(3 x-1)^{2}=12$$

Give a short written answer. The graphs of \(f(x)=x^{n}\) for \(n=3,5,7, \ldots\) resemble each other. As \(n\) gets larger, what happens to the graph?

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{3}+2 x^{2}-17 x-10 ; \quad x+5$$

Solve each problem. Volume of a Box \(\quad\) A piece of cardboard is twice as long as it is wide. It is to be made into a box with an open top by cutting 2 -inch squares from each corner and folding up the sides. Let \(x\) represent the width of the original piece of cardboard. (a) Represent the length of the original piece of cardboard in terms of \(x .\) (b) What will be the dimensions of the bottom rectangular base of the box? Give the restrictions on \(x .\) (c) Determine a function \(V\) that represents the volume of the box in terms of \(x\) (d) For what dimensions of the bottom of the box will the volume be 320 cubic inches? Determine analytically and support graphically. (e) Determine graphically (to the nearest tenth of an inch) the values of \(x\) if the box is to have a volume between 400 and 500 cubic inches. (IMAGE CAN'T COPY)

Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$1+\sqrt{2}, 1-\sqrt{2}, \text { and } 3$$

Write each number in simplest form, without a negative radicand. $$\sqrt{-169}$$

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. $$\begin{aligned} &9 x^{4}+35 x^{2}-4=0\\\ &[-3,3] \text { by }[-10,100] \end{aligned}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$(4 x+1)^{2}=20$$