Chapter 3

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{4}+4 x^{3}+2 x^{2}+9 x+4 ; \quad x+4$$

Solve each problem. Volume of a Box A piece of sheet metal is 2.5 times as long as it is wide. It is to be made into a box with an open top by cutting 3 -inch squares from each corner and folding up the sides, as shown at the top of the next page. Let \(x\) represent the width of the original piece of sheet metal. (a) Represent the length of the original piece of sheet metal in terms of \(x .\) (b) What are the restrictions on \(x ?\) (c) Determine a function \(V\) that represents the volume of the box in terms of \(x\) (d) For what values of \(x\) (that is, original widths) will the volume of the box be between 600 and 800 cubic inches? Determine the answer graphically, and give values to the nearest tenth of an inch. (IMAGE CAN'T COPY)

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

Write each number in simplest form, without a negative radicand. $$-\sqrt{-400}$$

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. $$\begin{aligned} &x^{3}-x^{2}-64 x+64=0\\\ &[-10,10] \text { by }[-300,300] \end{aligned}$$

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=3 x^{3}-11 x^{2}-20 x+3 ; \quad x-5$$

Solve each problem. Radius of a Can A can of garbanzo beans has surface. area 54.19 square inches. Its height is 4.25 inches. What is the radius of the circular top? See the figure. (Hint: The surface area consists of the circular top and bottom and a rectangle that represents the side cut open vertically and unrolled.) (IMAGE CAN'T COPY)

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

Write each number in simplest form, without a negative radicand. $$-\sqrt{-225}$$

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. $$\begin{aligned} &x^{3}+6 x^{2}-100 x-600=0\\\ &[-15,15] \text { by }[-1000,300] \end{aligned}$$

Solve each problem. Dimensions of a Cereal Box The volume of a 10 -ounce box of cereal is 182.742 cubic inches. The width of the box is 3.1875 inches less than the length, and its depth is 2.3125 inches. Find the length and width of the box to the nearest thousandth.

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{4}-3 x^{3}-5 x^{2}+2 x-16 ; \quad x-3$$

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

Write each number in simplest form, without a negative radicand. $$-\sqrt{-39}$$

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

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=2 x+24$$

Each function is graphed in a window that results in hidden behavior. Experiment with various windows to locate the extreme points on the graph of the function. Round values to the nearest hundredth. (GRAPH CANT COPY) $$y=\frac{1}{3} x^{3}-\frac{5}{2} x^{2}+6 x-1$$

Solve each problem. Radius Covered by a Circular Lawn Sprinkler A square lawn has area 800 square feet. A sprinkler placed at the center of the lawn sprays water in a circular pattern that just covers the lawn. What is the radius of the circle?

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{4}-3 x^{3}-4 x^{2}+12 x ; \quad x-2$$

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. $$P(x)=x^{2}-10 x+21$$

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

Write each number in simplest form, without a negative radicand. $$-\sqrt{-95}$$

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. $$\begin{aligned} &-5 x^{3}+13 x^{2}+6 x=0\\\ &[-4,4] \text { by }[-2,30] \end{aligned}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x^{2}=3 x+18$$

Each function is graphed in a window that results in hidden behavior. Experiment with various windows to locate the extreme points on the graph of the function. Round values to the nearest hundredth. (GRAPH CANT COPY) $$y=-\frac{1}{3} x^{3}-\frac{9}{2} x^{2}-20 x-\frac{59}{3}$$

Solve each problem. Height of a Kite \(\quad\) A kite is flying on 50 feet of string. How high is it above the ground if its height is 10 feet more than the horizontal distance from the person flying it? Assume that the string is being held at ground level. (IMAGE CAN'T COPY)

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=2 x^{4}+3 x^{3}-5 x^{2}-18 x ; \quad x-2$$

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. $$P(x)=x^{2}-2 x+3$$

Find a polynomial function \(P(x)\) having leading coefficient \(1,\) least possible degree, real coefficients, and the given zeros. $$-1 \text{ and } 6-3 i$$

Write each number in simplest form, without a negative radicand. $$5+\sqrt{-4}$$

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. $$\begin{aligned} &x^{3}+x^{2}-7 x-7=0\\\ &[-10,10] \text { by }[-20,20] \end{aligned}$$

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

Each function is graphed in a window that results in hidden behavior. Experiment with various windows to locate the extreme points on the graph of the function. Round values to the nearest hundredth. (GRAPH CANT COPY) $$y=-x^{3}-11 x^{2}-40 x-50$$

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{3}+2 x^{2}-3 ; \quad x-1$$

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. $$y=-x^{2}+4 x-2$$

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

Solve each problem. A raised wooden walkway is being constructed through a wetland. The walkway will have the shape of a right triangle with one leg 700 yards longer than the other and the hypotenuse 100 yards longer than the longer leg. Find the total length of the walkway.

Write each number in simplest form, without a negative radicand. $$-7+\sqrt{-100}$$

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. $$\begin{aligned} &x^{3}+3 x^{2}-19 x-57=0\\\ &[-10,10] \text { by }[-100,50] \end{aligned}$$

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

Each function is graphed in a window that results in hidden behavior. Experiment with various windows to locate the extreme points on the graph of the function. Round values to the nearest hundredth. (GRAPH CANT COPY) $$y=2 x^{3}-3.3 x^{2}+1.8 x+3$$

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=x^{3}-2 x^{2}-9 ; \quad x-3$$

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. $$y=-x^{2}+2 x+1$$

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

Write each number in simplest form, without a negative radicand. $$9-\sqrt{-50}$$

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. $$\begin{aligned} &-3 x^{3}-x^{2}+6 x=0\\\ &[-4,4] \text { by }[-10,10] \end{aligned}$$

Solve each equation. For equations with real solutions, support your answers graphically. $$x(14 x+1)=3$$

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) $$y=x^{2}-4.25 x+4.515$$

Find each quotient when \(P(x)\) is divided by the binomial following it. $$P(x)=-2 x^{3}-x-2 ; \quad x+1$$

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. $$P(x)=2 x^{2}-4 x+5$$