Chapter 6

For Exercises 32 and \(33,\) use the following information. The average height (in inches) for boys ages 1 to 20 can be modeled by the equation \(B(x)=-0.001 x^{4}+0.04 x^{3}-0.56 x^{2}+5.5 x+25\) , where \(x\) is the age (in years). The average height for girls ages 1 to 20 is modeled by the equation \(G(x)=-0.0002 x^{4}+0.006 x^{3}-0.14 x^{2}+3.7 x+26\) . Graph both equations by making a table of values. Use \(x=\\{0,2,4,6,8,10,\) \(12,14,16,18,20 \\}\) as the domain. Round values to the nearest inch.

A restaurant orders spaghetti sauce in cylindrical metal cans. The volume of each can is about 160\(\pi\) cubic inches, and the height of the can is 6 inches more than the radius. Write a polynomial equation that represents the volume of a can. Use the formula for the volume of a cylinder, \(V=\pi r^{2} h .\)

Find all of the zeros of each function. \(h(x)=x^{4}-15 x^{3}+70 x^{2}-70 x-156\)

Solve each equation. $$ x^{4}+6 x^{2}-27=0 $$

Simplify. $$ (6-z)(6+z) $$

Simplify. $$ \frac{x^{4}+x^{2}-3 x+5}{x^{2}+2} $$

Simplify. Assume that no variable equals 0. $$ \frac{12 x^{-3} y^{-2} z^{-8}}{30 x^{-6} y^{-4} z^{-1}} $$

ENGINEERING. For Exercises 32 and \(33,\) use the following information. When a certain type of plastic is cut into sections, the length of each section determines its strength. The function \(f(x)=x^{4}-14 x^{3}+69 x^{2}-140 x+100\) can describe the relative strength of a section of length \(x\) feet. Sections of plastic \(x\) feet long, where \(f(x)=0,\) are extremely weak. After testing the plastic, engineers discovered that sections 5 feet long were extremely weak. Are there other lengths of plastic that are extremely weak? Explain your reasoning.

A restaurant orders spaghetti sauce in cylindrical metal cans. The volume of each can is about 160\(\pi\) cubic inches, and the height of the can is 6 inches more than the radius. What are the possible values of \(r ?\) Which values are reasonable here?

Write a polynomial function of least degree with integral coefficients that has the given zeros. \(-4,1,5\)

Solve each equation. $$ x^{3}+64=0 $$

Simplify. $$ (3 x+8)(2 x+6) $$

Simplify. $$ \frac{2 x^{4}+3 x^{3}-2 x^{2}-3 x-6}{2 x+3} $$

Simplify. Assume that no variable equals 0. $$ \left(\frac{x}{y^{-1}}\right)^{-2} $$

Find values of \(k\) so that each remainder is \(3 .\) $$ \left(x^{2}-x+k\right) \div(x-1) $$

Use a graphing calculator to estimate the \(x\) -coordinates at which the maxima and minima of each function occur. Round to the nearest hundredth. $$ f(x)=x^{3}+x^{2}-7 x-3 $$

A restaurant orders spaghetti sauce in cylindrical metal cans. The volume of each can is about 160\(\pi\) cubic inches, and the height of the can is 6 inches more than the radius. Find the dimensions of the can.

Write a polynomial function of least degree with integral coefficients that has the given zeros. \(-2,2,4,6\)

Solve each equation. $$ 27 x^{3}+1=0 $$

Simplify. $$ (4 y-6)(2 y+7) $$

Simplify. $$ \frac{6 x^{4}+5 x^{3}+x^{2}-3 x+1}{3 x+1} $$

Simplify. Assume that no variable equals 0. $$ \left(\frac{v}{w^{-2}}\right)^{-3} $$

Find values of \(k\) so that each remainder is \(3 .\) $$ \left(x^{2}+k x-17\right) \div(x-2) $$

Use a graphing calculator to estimate the \(x\) -coordinates at which the maxima and minima of each function occur. Round to the nearest hundredth. $$ f(x)=-x^{3}+6 x^{2}-6 x-5 $$

An amusement park owner wants to add a new wilderness water ride that includes a mountain that is shaped roughly like a square pyramid. Before building the new attraction, engineers must build and test a scale model. If the height of the scale model is 9 inches less than its length, write a polynomial function that describes the volume of the model in terms of its length. Use the formula for the volume of a pyramid, \(V=\frac{1}{3} B h\).

Write a polynomial function of least degree with integral coefficients that has the given zeros. \(4 i, 3,-3\)

Solve each equation. $$ 8 x^{3}-27=0 $$

For a moving object with mass \(m\) in kilograms, the kinetic energy \(K E\) in joules is given by the function \(K E(v)=\frac{1}{2} m v^{2},\) where \(v\) represents the speed of the object in meters per second. Find the kinetic energy of an all-terrain vehicle with a mass of 171 kilograms moving at a speed of 11 meters/ second.

Simplify. $$ (3 b-c)^{3} $$

A magician gives these instructions to a volunteer. \(\cdot\) Choose a number and multiply it by \(4 .\) \(\cdot\) Then add the sum of your number and 15 to the product you found. \(\cdot\) Now divide by the sum of your number and \(3 .\) What number will the volunteer always have at the end? Explain.

Simplify. Assume that no variable equals 0. $$ \left(\frac{8 a^{3} b^{2}}{16 a^{2} b^{3}}\right)^{4} $$

Find values of \(k\) so that each remainder is \(3 .\) $$ \left(x^{2}+5 x+7\right) \div(x+k) $$

Use a graphing calculator to estimate the \(x\) -coordinates at which the maxima and minima of each function occur. Round to the nearest hundredth. $$ f(x)=-x^{4}+3 x^{2}-8 $$

Write a polynomial function of least degree with integral coefficients that has the given zeros. \(2 i, 3 i, 1\)

DESIGN. For Exercises \(36-38,\) use the following information. Jill is designing a picture frame for an art project. She plans to have a square piece of glass in the center and surround it with a decorated ceramic frame, which will also be a square. The dimensions of the glass and frame are shown in the diagram at the right. jill determines that she needs 27 square inches of material for the frame. Write a polynomial equation that models the area of the frame.

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=x^{4}-7 x^{3}+8 x-6\)

Simplify. $$ \left(x^{2}+x y+y^{2}\right)(x-y) $$

Simplify. Assume that no variable equals 0. $$ \left(\frac{6 x^{2} y^{4}}{3 x^{4} y^{3}}\right)^{3} $$

Find values of \(k\) so that each remainder is \(3 .\) $$ \left(x^{3}+4 x^{2}+x+k\right) \div(x+2) $$

Use a graphing calculator to estimate the \(x\) -coordinates at which the maxima and minima of each function occur. Round to the nearest hundredth. $$ f(x)=3 x^{4}-7 x^{3}+4 x-5 $$

Write a polynomial function of least degree with integral coefficients that has the given zeros. \(9,1+2 i\)

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=7 x^{2}-9 x+10\)

PERSONAL FINANCE. Toshiro has \(\$ 850\) to invest. He can invest in a savings account that has an annual interest rate of 1.7\(\%\) , and he can invest in a money market account that pays about 3.5\(\%\) per year. Write a polynomial to represent the amount of interest he will earn in 1 year if he invests \(x\) dollars in the savings account and the rest in the money market account.

The number of sports magazines sold can be estimated by \(n=\frac{3500 a^{2}}{a^{2}+100},\) where \(a\) is the amount of money spent on advertising in hundreds of dollars and \(n\) is the number of subscriptions sold. About how many subscriptions will be sold if \(\$ 1500\) is spent on advertising?

Simplify. Assume that no variable equals 0. $$ \left(\frac{4 x^{-3} y^{2}}{x y^{-5}}\right)^{-2} $$

PERSONAL FINANCE For Exercises \(38-41,\) use the following information. Zach has purchased some home theater equipment for \(\$ 2000,\) which he is financing through the store. He plans to pay \(\$ 340\) per month and wants to have the balance paid off after six months. The formula \(B(x)=2000 x^{6}-\) 340\(\left(x^{5}+x^{4}+x^{3}+x^{2}+x+1\right)\) represents his balance after six months if \(x\) represents 1 plus the monthly interest rate (expressed as a decimal). Find his balance after 6 months if the annual interest rate is 12\(\% .\) (Hint: The monthly interest rate is the annual rate divided by \(12,\) so \(x=1.01 . )\)

OPEN ENDED Sketch a graph of a function that has one relative maximum point and two relative minimum points.

Find all of the zeros of \(f(x)=x^{3}-2 x^{2}+3\) and \(g(x)=2 x^{3}-7 x^{2}+2 x+3\)

Write a polynomial function of least degree with integral coefficients that has the given zeros. \(6,2+2 i\)

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=\frac{1}{2} x^{4}-2 x^{2}+4\)