Chapter 6

E-SALES. For Exercises 38 and \(39,\) use the following information. A small online retailer estimates that the cost, in dollars, associated with selling \(x\) units of a particular product is given by the expression \(0.001 x^{2}+5 x+500 .\) The revenue from selling \(x\) units is given by 10\(x\) . Write a polynomial to represent the profit generated by the product.

For Exercises \(38-40,\) suppose an object moves in a straight line so that, after \(t\) seconds, it is \(t^{3}+t^{2}+6 t\) feet from its starting point. Find the distance the object travels between the times \(t=2\) and \(t=x\) where \(x > 2\)

If \(2^{r+5}=2^{2 r-1},\) what is the value of \(r ?\)

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 9.6\(\%\)

GEOMETRY The width of a rectangular prism is \(w\) centimeters. The height is 2 centimeters less than the width. The length is 4 centimeters more than the width. If the volume of the prism is 8 times the measure of the length, find the dimensions of the prism.

Find \(p(4)\) and \(p(-2)\) for each function. \(p(x)=\frac{1}{8} x^{3}-\frac{1}{4} x^{2}-\frac{1}{2} x+5\)

E-SALES. For Exercises 38 and \(39,\) use the following information. A small online retailer estimates that the cost, in dollars, associated with selling \(x\) units of a particular product is given by the expression \(0.001 x^{2}+5 x+500 .\) The revenue from selling \(x\) units is given by 10\(x\) . Find the profit from sales of 1850 units.

What value of \(r\) makes \(y^{28}=y^{3 r} \cdot y^{7}\) true?

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). How would the formula change if Zach wanted to pay the balance in five months?

For Exercises \(39-41,\) sketch a graph of each polynomial. odd-degree polynomial function with one relative maximum and one relative minimum; the leading coefficient is negative

A computer manufacturer determines that for each employee the profit for producing \(x\) computers per day is \(P(x)=-0.006 x^{4}+0.15 x^{3}-0.05 x^{2}-1.8 x\) Approximate all real zeros to the nearest tenth by graphing the function using a graphing calculator.

Find the factorization of \(3 x^{2}+x-2\)

If \(p(x)=3 x^{2}-2 x+5\) and \(r(x)=x^{3}+x+1,\) find each value. 2\([p(x+4)]\)

Simplify \(\left(c^{2}-6 c d-2 d^{2}\right)+\left(7 c^{2}-c d+8 d^{2}\right)-\left(-c^{2}+5 c d-d^{2}\right)\)

For Exercises \(38-40,\) suppose an object moves in a straight line so that, after \(t\) seconds, it is \(t^{3}+t^{2}+6 t\) feet from its starting point. Find a simplified expression for the average speed of the object between times \(t=2\) and \(t=x\)

INCOME In \(2003,\) the population of Texas was about \(2.21 \times 10^{7}\) . The personal income for the state that year was about \(6.43 \times 10^{11}\) dollars. What was the average personal income?

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). Suppose he finances his purchase at 10.8\(\%\) and plans to pay \(\$ 410\) every month. Will his balance be paid in full after five months?

For Exercises \(39-41,\) sketch a graph of each polynomial. odd-degree polynomial function with three relative maxima and three relative minima; the leftmost points are negative

A computer manufacturer determines that for each employee the profit for producing \(x\) computers per day is \(P(x)=-0.006 x^{4}+0.15 x^{3}-0.05 x^{2}-1.8 x\) What is the meaning of the roots in this problem?

What are the factors of \(2 y^{2}+9 y+4 ?\)

If \(p(x)=3 x^{2}-2 x+5\) and \(r(x)=x^{3}+x+1,\) find each value. \(r(x+1)-r\left(x^{2}\right)\)

Find the product of \(x^{2}+6 x-5\) and \(-3 x+2\)

OPEN ENDED Write a quotient of two polynomials such that the remainder is \(5 .\)

RESEARCH Use the Internet or other source to find the masses of Earth and the Sun. About how many times as large as Earth is the Sun?

OPEN ENDED. Give an example of a polynomial function that has a remainder of 5 when divided by \(x-4 .\)

If \(k\) and 2\(k\) are zeros of \(f(x)=x^{3}+4 x^{2}+$$9 k x-90,\) find \(k\) and all three zeros of \(f(x) .\)

The space shuttle has an external tank for the fuel that the main engines need for the launch. This tank is shaped like a capsule, a cylinder with a hemispherical dome at either end. The cylindrical part of the tank has an approximate volume of 336\(\pi\) cubic meters and a height of 17 meters more than the radius of the tank. \(\left(\text {Hint: } V(r)=\pi r^{2} h\right)\). Write an equation that represents the volume of the cylinder.

Factor completely. If the polynomial is not factorable, write prime. $$ 3 n^{2}+21 n-24 $$

Simplify. $$ \left(4 x^{2}-3 y^{2}+5 x y\right)-\left(8 x y+3 y^{2}\right) $$

OPEN ENDED Write an example that illustrates a property of powers. Then use multiplication or division to explain why it is true.

The space shuttle has an external tank for the fuel that the main engines need for the launch. This tank is shaped like a capsule, a cylinder with a hemispherical dome at either end. The cylindrical part of the tank has an approximate volume of 336\(\pi\) cubic meters and a height of 17 meters more than the radius of the tank. \(\left(\text {Hint: } V(r)=\pi r^{2} h\right)\). What are the dimensions of the cylindrical part of the tank?

Factor completely. If the polynomial is not factorable, write prime. $$ y^{4}-z^{2} $$

Simplify. $$ \left(10 x^{2}-3 x y+4 y^{2}\right)-\left(3 x^{2}+5 x y\right) $$

FIND THE ERROR. Alejandra and Kyle both simplified \(\frac{2 a^{2} b}{\left(-2 a^{3} b\right)^{-2}} .\) Who is correct? Explain your reasoning. $$ \begin{array}{l}{\text { Alejandra }} \\ {\begin{aligned} \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{2}} &=\left(2 a^{2} b\right)\left(-2 a b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right)(-2)^{2} a^{2}\left(b^{3}\right)^{2} \\ &=\left(2 a^{2} b\right) 2^{2} a^{2} b^{6} \\\ &=8 a^{4} b^{7} \end{aligned}}\end{array} $$ $$ \begin{aligned} & \text { Kyle } \\ \frac{2 a^{2} b}{\left(-2 a b^{3}\right)^{-2}} &=\frac{2 a^{2} b}{(-2)^{2} a\left(b^{3}\right)^{-2}} \\\ &=\frac{2 a^{2} b}{4 a b^{-6}} \\ &=\frac{2 a^{2} b b^{6}}{4 a} \\ &=\frac{a b^{7}}{2} \end{aligned} $$

CHALLENGE. Consider the polynomial \(f(x)=a x^{4}+b x^{3}+c x^{2}+d x+e,\) where \(a+b+c+d+e=0 .\) Show that this polynomial is divisible by \(x-1\)

Which of the following is a zero of the function \(f(x)=12 x^{5}-5 x^{3}+2 x-9 ?\) A. \(-6\) B. \(\frac{3}{8}\) C. \(-\frac{2}{3}\) D. 1

Antonio is preparing to make an ice sculpture. He has a block of ice that he wants to reduce in size by shaving off the same amount from the length, width, and height. He wants to reduce the volume of the ice block to 24 cubic feet. Write a polynomial equation to model this situation.

Factor completely. If the polynomial is not factorable, write prime. $$ 16 a^{2}+25 b^{2} $$

Simplify. $$ \frac{3}{4} x^{2}\left(8 x+12 y-16 x y^{2}\right) $$

REASONING. Determine whether \(x^{y} \cdot x^{z}=x^{y z}\) is sometimes, always, or never true. Explain your reasoning.

REVIEW Mandy went shopping. She spent two-fifths of her money in the first store. She spent three-fifths of what she had left in the next store. In the last store she visited, she spent three-fourths of the money she had left. When she finished shopping, Mandy had \(\$ 6 .\) How much money in dollars did Mandy have when she started shopping? $$ \begin{array}{lll}{\mathbf{F}} & {\$ 16} & {\mathbf{H}} & {\$ 100} \\\ {\mathbf{G}} & {\$ 56} & {\mathbf{J}} & {\$ 106}\end{array} $$

window is in the shape of an equilateral triangle. Each side of the triangle is 8 feet long. The window is divided in half by a support from one vertex to the midpoint of the side of the triangle opposite the vertex. Approximately how long is the support? F. 5.7 ft G. 6.9 ft H. 11.3 ft J. 13.9 ft

Antonio is preparing to make an ice sculpture. He has a block of ice that he wants to reduce in size by shaving off the same amount from the length, width, and height. He wants to reduce the volume of the ice block to 24 cubic feet. How much should he take from each dimension?

Factor completely. If the polynomial is not factorable, write prime. $$ 3 x^{2}-27 y^{2} $$

Simplify. $$ \frac{1}{2} a^{3}\left(4 a-6 b+8 a b^{4}\right) $$

CHALLENGE Determine which is greater, \(100^{10}\) or \(10^{100} .\) Explain.

If \(p(x)=2 x^{2}-5 x+4\) and \(r(x)=3 x^{3}-x^{2}-2,\) find each value. $$ r(2 a) $$

Given a function and one of its zeros, find all of the zeros of the function. \(g(x)=x^{3}+4 x^{2}-27 x-90 ;-3\)

The number of regions formed by connecting \(n\) points of a circle can be described by the function \(f(n)=\frac{1}{24}\left(n^{4}-6 n^{3}+23 n^{2}-18 n+24\right) .\) What is the degree of this polynomial function?

Factor completely. If the polynomial is not factorable, write prime. $$ x^{4}-81 $$