Chapter 6

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{6 x^{2}+13 x+6}{6-5 x-6 x^{2}} $$

When dividing a polynomial by a binomial of the form \(x-k\) synthetic division is considered to be faster than long division. Explain why.

For each expression in part (a), perform the indicated operations and then simplify, if possible. Solve equation in part (b) and check the result. a. \(\frac{11}{12}-\frac{3}{2 x}+\frac{4}{x}\) b. \(\frac{11}{12}-\frac{3}{2 x}=\frac{4}{x}\)

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

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{6}{b^{2}-9} \cdot \frac{b+3}{2 b+4}$$

Perform the operations and simplify. $$ \begin{aligned} &\text { Let } s(x)=\frac{x^{2}-16}{x^{2}-25} \text { and } f(x)=\frac{5 x+20}{10 x^{2}-50 x}\\\ &\text { Find } s(x) \div f(x) \end{aligned} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{x^{2}-6 x+9}{81-x^{4}} $$

Let \(P(x)=x^{3}-6 x^{2}-9 x+4 .\) You now know two ways to find \(P(6) .\) What are they? Which method do you prefer?

For each expression in part (a), perform the indicated operations and then simplify, if possible. Solve equation in part (b) and check the result. a. \(\frac{1}{6 x}-\frac{2}{x-6}\) b. \(\frac{1}{6 x}=\frac{2}{x-6}\)

Simplify each expression. $$ \left(\frac{\frac{2}{b}+\frac{1}{2 b}}{b+\frac{b}{2}}\right)+\left(\frac{b-\frac{b-3}{3}}{\frac{4}{9}+\frac{2}{3 b}}\right) $$

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{3 a^{2}-22 a+7}{a-a^{2}} \cdot \frac{8 a^{2}-8 a}{a^{2}+a-56}$$

Solve each problem by writing a variation model. Organ Pipes. The frequency of vibration of air in an organ pipe is inversely proportional to the length of the pipe. If a pipe 2 feet long vibrates 256 times per second, how many times per second will a 6 -foot pipe vibrate?

Perform the operations and simplify. $$ \begin{aligned} &\text { Let } g(x)=\frac{x^{2}-9}{x^{2}-49} \text { and } h(x)=\frac{9 x^{2}+27 x}{3 x+21}\\\ &\text { Find } g(x) \div h(x) \end{aligned} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{y^{2}-2 y+1}{1-y^{4}} $$

Explain the factor theorem.

For each expression in part (a), perform the indicated operations and then simplify, if possible. Solve equation in part (b) and check the result. a. \(\frac{m}{m-2}-\frac{1}{m-3}\) b. \(\frac{m}{m-2}-\frac{1}{m-3}=1\)

Solve each problem by writing a variation model. Gas Pressure. Under constant temperature, the volume occupied by a gas varies inversely to the pressure applied. If the gas occupies a volume of 20 cubic inches under a pressure of 6 pounds per square inch, find the volume when the gas is subjected to a pressure of 10 pounds per square inch.

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{4 a}{a-5}+a$$

Look Alikes . . . a. \(\frac{t^{2}+9 t+20}{9 t+36} \cdot \frac{9 t+45}{t+4}\) b. \(\frac{t^{2}+9 t+20}{9 t+36} \div \frac{9 t+45}{t+4}\)

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{16 p^{3} q^{2}}{24 p q^{8}} $$

This section includes a feature entitled Using Your Calculator: Approximating Zeros of Polynomials. What is a zero of a polynomial?

For each expression in part (a), perform the indicated operations and then simplify, if possible. Solve equation in part (b) and check the result. a. \(\frac{a^{2}+1}{a^{2}-a}-\frac{a}{a-1}\) b. \(\frac{a^{2}+1}{a^{2}-a}-\frac{a}{a-1}=\frac{1}{a}\)

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{10 z}{z+4}+z$$

Look Alikes . . . a. \(\frac{a^{2}-5 a+6}{2 a-4} \cdot \frac{2 a-6}{a-2}\) b. \(\frac{a^{2}-5 a+6}{2 a-4} \div \frac{2 a-6}{a-2}\)

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{30 a^{3} b^{15}}{18 a^{9} b^{10}} $$

Solve each equation. $$ |3 x-7|+8=22 $$

Solve each problem by writing a variation model. Trucking costs. The costs of a trucking company vary jointly as the number of trucks in service and the number of hours they are used. When 4 trucks are used for 6 hours each, the costs are \(\$ 1,800 .\) Find the costs of using 10 trucks, each for 12 hours.

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{2 a+1}{3 a-2}-\frac{a-4}{2-3 a}$$

Look Alikes . . . a. \(\frac{4 r-8}{5 r} \div \frac{5 r-10}{4 r^{2}}\) b. \(\frac{4 r-8}{5 r} \cdot \frac{5 r-10}{4 r^{2}}\)

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{t^{3}-5 t^{2}+6 t}{9 t-t^{3}} $$

Solve each equation. $$ |75-x|=-1 $$

The focal length, \(f,\) of a lens is given by the lensmaker's formula, $$\frac{1}{f}=0.6\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)$$ where \(f\) is the focal length of the lens and \(r_{1}\) and \(r_{2}\) are the radii of the two circular surfaces. Solve the formula for \(f\).

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{2 x+1}{x^{4}-81}+\frac{2-x}{x^{4}-81}$$

Look Alikes . . . a. \(\frac{3 d^{2}+6}{4} \div \frac{4 d^{2}+8}{3}\) b. \(\frac{3 d^{2}+6}{4} \cdot \frac{4 d^{2}+8}{3}\)

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{a^{4}-27 a}{36 a-4 a^{3}} $$

Solve each equation. $$ 6-3|10 x+5|=6 $$

Accounting. As a piece of equipment gets older, its value usually lessens. One way to calculate depreciation is to use the formula $$V=C-\left(\frac{C-S}{L}\right) N$$ where \(V\) denotes the value of the equipment at the end of year \(N, L\) is its useful lifetime (in years), \(C\) is its cost new, and \(S\) is its salvage value at the end of its useful life. a. Solve the formula for \(L\) b. Determine what an accountant considered the useful lifetime of a forklift that cost \(\$ 25,000\) new, was worth \(\$ 13,000\) after 4 years, and has a salvage value of \(\$ 1,000\).

Solve each problem by writing a variation model. Electronics. The voltage (in volts) measured across a resistor is directly proportional to the current (in amperes) flowing through the resistor. The constant of variation is the resistance (in ohms). If 6 volts is measured across a resistor carrying a current of 2 amperes, find the resistance.

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{x^{2}+x}{3 x-15} \div \frac{(x+1)^{2}}{6 x-30}$$

Simplify each expression. \(2\left(x^{2}+4 x-1\right)+3\left(2 x^{2}-2 x+2\right)\)

Suppose the cost (in dollars) of removing \(p \%\) of the pollution in a river is given by the rational function \(f(p)=\frac{50,000 p}{100-p}\) where \(0 \leq p<100\) Find the cost of removing each percent of pollution. a. \(50 \%\) b. \(80 \%\)

Solve each equation. $$ |2-x|=|3 x+2| $$

The equation $$a=\frac{9.8 m_{2}-f}{m_{2}+m_{1}}$$ models the system shown, where \(a\) is the acceleration of the suspended block, \(m_{1}\) and \(m_{2}\) are the masses of the blocks, and \(f\) is the friction force. Solve for \(m_{2}\).

Solve each problem by writing a variation model. Electronics. The power (in watts) lost in a resistor (in the form of heat) varies directly as the square of the current (in amperes) passing through it. The constant of proportionality is the resistance (in ohms). What power is lost in a 5 -ohm resistor carrying a 3 -ampere current?

Perform the operations and simplify the result when possible. Be careful to apply the correct method, because these problems involve addition, subtraction, multiplication, and division of rational expressions. $$\frac{m}{m^{2}+5 m+6}-\frac{2}{m^{2}+3 m+2}$$

Simplify each expression. \(3\left(2 a^{2}-3 a+2\right)-4\left(2 a^{2}+4 a-7\right)\)

The average (mean) cost for a service club to publish a directory of its members is given by the rational function $$ f(x)=\frac{1.25 x+700}{x} $$ where \(x\) is the number of directories printed. Find the average cost per directory if a. 500 directories are printed. b. \(2,000\) directories are printed.

Suppose that \(P(x)=x^{100}-x^{99}+x^{98}-x^{97}+\cdots+x^{2}-x+1\). Find the remainder when \(P(x)\) is divided by \(x-1\)

Why is it necessary to check the solutions of a rational equation?

Solve each problem by writing a variation model. Structural Engineering. The deflection of a beam is inversely proportional to its width and the cube of its depth. If the deflection of a 4 -inch-wide by 4-inch-deep beam is 1.1 inches, find the deflection of a 2 -inch-wide by 8 -inch-deep beam positioned as in figure (a) below.