Chapter 6

Simplify. $$ \left(t^{5}-3 t^{2}-20\right)(t-2)^{-1} $$

Simplify. Assume that no variable equals 0. $$ \frac{2 c^{3} d\left(3 c^{2} d^{5}\right)}{30 c^{4} d^{2}} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ x^{4}+2 x^{3}+2 x^{2}-2 x-3 ; x+1 $$

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=-x^{4}+5 x^{2}-2 x-1 $$

Find all of the zeros of each function. \(h(x)=10 x^{3}-17 x^{2}-7 x+2\)

Find all of the zeros of each function. \(f(x)=x^{3}-7 x^{2}+25 x-175\)

Write each expression in quadratic form, if possible. $$ 11 n^{6}+44 n^{3} $$

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

Simplify. $$ 4 a\left(3 a^{2}+b\right) $$

Simplify. $$ \left(2 b^{3}+b^{2}-2 b+3\right)(b+1)^{-1} $$

Simplify. Assume that no variable equals 0. $$ \frac{-12 m^{4} n^{8}\left(m^{3} n^{2}\right)}{36 m^{3} n} $$

Given a polynomial and one of its factors, find the remaining factors of the polynomial. Some factors may not be binomials. $$ 16 x^{5}-32 x^{4}-81 x+162 ; x-2 $$

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=-x^{4}+x^{3}+8 x^{2}-3 $$

Find all of the zeros of each function. \(g(x)=48 x^{4}-52 x^{3}+13 x-3\)

Find all of the zeros of each function. \(g(x)=2 x^{3}-x^{2}+28 x+51\)

Write each expression in quadratic form, if possible. $$ 7 b^{5}-4 b^{3}+2 b $$

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

Simplify. $$ -5 a b^{2}\left(-3 a^{2} b+6 a^{3} b-3 a^{4} b^{4}\right) $$

Simplify. $$ \frac{x^{5}-7 x^{3}+x+1}{x+3} $$

Use synthetic substitution to show that \(x-8\) is a factor of \(x^{3}-4 x^{2}-29 x-24 .\) Then find any remaining factors.

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=x^{4}-9 x^{3}+25 x^{2}-24 x+6 $$

Find all of the zeros of each function. \(p(x)=x^{5}-2 x^{4}-12 x^{3}-12 x^{2}-13 x-10\)

Find all of the zeros of each function. \(q(x)=2 x^{3}-17 x^{2}+90 x-41\)

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

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

POPULATION. The population of Earth is about \(6.445 \times 10^{9} .\) The land surface area of Earth is \(1.483 \times 10^{8} \mathrm{km}^{2} .\) What is the population density for the land surface area of Earth?

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=2 x^{4}-4 x^{3}-2 x^{2}+3 x-5 $$

Find all of the zeros of each function. \(h(x)=9 x^{5}-94 x^{3}+27 x^{2}+40 x-12\)

Find all of the zeros of each function. \(h(x)=4 x^{4}+17 x^{2}+4\)

Write each expression in quadratic form, if possible. $$ 6 x^{\frac{2}{5}}-4 x^{\frac{1}{5}}-16 $$

Simplify. $$ (p+6)(p-4) $$

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

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

BOATING. For Exercises 30 and \(31,\) use the following information. A motor boat traveling against waves accelerates from a resting position. Suppose the speed of the boat in feet per second is given by the function \(f(t)=-0.04 t^{4}+0.8 t^{3}+0.5 t^{2}-t,\) where \(t\) is the time in seconds. Find the speed of the boat at \(1,2,\) and 3 seconds.

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=x^{5}+4 x^{4}-x^{3}-9 x^{2}+3 $$

The length of the cargo space in a sport-utility vehicle is 4 inches greater than the height of the space. The width is sixteen inches less than twice the height. The cargo space has a total volume of 55,296 cubic inches. Use a rectangular prism to model the cargo space. Write a polynomial function that represents the volume of the cargo space.

Find all of the zeros of each function. \(p(x)=x^{4}-9 x^{3}+24 x^{2}-6 x-40\)

Solve each equation. $$ x^{4}-34 x^{2}+225=0 $$

Simplify. $$ (a+6)(a+3) $$

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

Simplify. Assume that no variable equals 0. $$ 3 a\left(5 a^{2} b\right)\left(6 a b^{3}\right) $$

BOATING. For Exercises 30 and \(31,\) use the following information. A motor boat traveling against waves accelerates from a resting position. Suppose the speed of the boat in feet per second is given by the function \(f(t)=-0.04 t^{4}+0.8 t^{3}+0.5 t^{2}-t,\) where \(t\) is the time in seconds. It takes 6 seconds for the boat to travel between two buoys while it is accelerating. Use synthetic substitution to find \(f(6)\) and explain what this means.

For Exerises \(26-31\) , complete each of the following. a. Graph each funnction by making a table of values. b. Determine the consecutive integer values of \(x\) between which each real zero is located. C. Estimate the \(x\) -coordinates at which the relative maxima and relative minima occur. $$ f(x)=x^{5}-6 x^{4}+4 x^{3}+17 x^{2}-5 x-6 $$

The length of the cargo space in a sport-utility vehicle is 4 inches greater than the height of the space. The width is sixteen inches less than twice the height. The cargo space has a total volume of 55,296 cubic inches. Will a package 34 inches long, 44 inches wide, and 34 inches tall fit in the cargo space? Explain.

Find all of the zeros of each function. \(r(x)=x^{4}-6 x^{3}+12 x^{2}+6 x-13\)

Solve each equation. $$ x^{4}-15 x^{2}-16=0 $$

Simplify. $$ (b+5)(b-5) $$

Simplify. $$ \left(6 t^{3}+5 t^{2}+9\right) \div(2 t+3) $$

Simplify. Assume that no variable equals 0. $$ \frac{30 a^{-2} b^{-6}}{60 a^{-6} b^{-8}} $$

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. Show that \(x-5\) is a factor of the polynomial function.