Chapter 6

Perform each division. \(\frac{13 x+16 x^{4}+3 x^{2}+3}{3+4 x}\)

Simplify each rational expression. $$ \frac{x^{2}+2 x+1}{x^{2}+4 x+3} $$

Simplify each complex fraction. $$ \frac{\frac{2 x-t x+2 y-t y}{x^{2}+2 x y+y^{2}}}{\frac{t^{3}-8}{15 x+15 y}} $$

Solve each formula for the specified variable. \(\frac{P_{1} V_{1}}{T_{1}}=\frac{P_{2} V_{2}}{T_{2}}\) for \(T_{2}\) (from chemistry)

Add or subtract, and then simplify, if possible. See Example 4. $$\frac{b-3}{b+4}-\frac{b+2}{b-4}$$

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ A=k r^{3} $$

Write a shared-work problem that can be modeled by the equation $$ \frac{x}{3}+\frac{x}{4}=1 $$

Perform each division. \(\frac{4 x^{3}-12 x^{2}+17 x-12}{2 x-3}\)

Simplify each complex fraction. $$ \frac{\frac{8}{x}+\frac{x}{8}}{\frac{x}{8}-\frac{8}{x}} $$

Solve each formula for the specified variable. \(P=\frac{Q_{1}}{Q_{2}-Q_{1}}\) for \(Q_{1}\) (from refrigeration/heating)

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ b=\frac{k}{h} $$

Simplify each expression. Write answers using positive exponents. $$ \left(\frac{m^{10}}{n}\right)^{8} $$

Perform each division. Divide \(8 a^{3}+1\) by \(2 a+1\)

Divide, and then simplify, if possible. See Objective 3. $$ \frac{6}{11} \div \frac{36}{55} $$

Simplify each rational expression. $$ \frac{3 d^{2}+13 d+4}{3 d^{2}+7 d+2} $$

Simplify each complex fraction. $$ \frac{\frac{x}{7}-\frac{7}{x}}{\frac{1}{7}+\frac{1}{x}} $$

Solve each formula for the specified variable. \(S=\frac{a-\ell r}{1-r}\) for \(r\) (from mathematics)

Add or subtract, and then simplify, if possible. See Example 5 $$2-\frac{1}{x+1}$$

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ d=\frac{k}{W^{4}} $$

Simplify each expression. Write answers using positive exponents. $$ \left(\frac{g^{20}}{t^{30}}\right)^{-4} $$

Perform each division. Divide \(27 a^{3}-8\) by \(3 a-2\)

Divide, and then simplify, if possible. See Objective 3. $$ \frac{17}{12} \div \frac{34}{3} $$

Simplify each rational expression. $$ \frac{10 r^{2}+17 r+3}{2 r^{2}+17 r+21} $$

Simplify each complex fraction. $$ -\frac{a}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}} $$

Solve each formula for the specified variable. \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) for \(R\) (from electronics)

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ U=k r s^{2} t $$

Simplify each expression. Write answers using positive exponents. $$ -w^{-2} $$

Perform each division. Divide \(15 a^{3}-29 a^{2}+16\) by \(3 a-4\)

Divide, and then simplify, if possible. See Objective 3. $$ \frac{12}{5} \div \frac{24}{45} $$

Simplify each rational expression. $$ \frac{2 h^{2}+9 h-5}{4 h^{2}-4 h+1} $$

Let \(Q(x)=x^{4}-3 x^{3}+2 x^{2}+x-3 .\) Evaluate \(Q(x)\) by substituting the given value of \(x\) into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. See Example 4. $$ Q(1) $$

Simplify each complex fraction. $$ -\frac{m}{\frac{1}{m}-\frac{1}{n}+\frac{1}{t}} $$

Solve each formula for the specified variable. \(\frac{x}{a}+\frac{y}{b}=1\) for \(a\) (from mathematics)

Add or subtract, and then simplify, if possible. See Example 5 $$x-\frac{3 x}{3 x-2}$$

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ L=k m n $$

Simplify each expression. Write answers using positive exponents. $$ -3 s^{0} t $$

Perform each division. Divide \(15 c^{3}-19 c^{2}+4\) by \(5 c+2\)

Divide, and then simplify, if possible. See Objective 3. $$ \frac{18}{7} \div \frac{54}{35} $$

Simplify each rational expression. $$ \frac{6 x^{2}+x-2}{8 x^{2}+2 x-3} $$

Let \(Q(x)=x^{4}-3 x^{3}+2 x^{2}+x-3 .\) Evaluate \(Q(x)\) by substituting the given value of \(x\) into the polynomial and simplifying. Then evaluate the polynomial by using the remainder theorem and synthetic division. See Example 4. $$ Q(2) $$

Simplify each complex fraction. $$ \frac{\frac{1}{a+1}+1}{\frac{3}{a-1}+1} $$

Solve each formula for the specified variable. \(\frac{E}{e}=\frac{R+r}{r}\) for \(r\) (from engineering)

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ P=\frac{k m}{n} $$

Simplify each expression. Write answers using positive exponents. $$ -\frac{4 x^{-9} \cdot x^{-3}}{x^{-12}} $$

Perform each division. Divide \(7 x^{2}-x+x^{4}+5 x^{3}-12\) by \(x^{2}-3+2 x\)

Divide, and then simplify, if possible. See Example 6. $$ \frac{22 x^{3}}{y^{2}} \div \frac{33 x^{9}}{y^{7}} $$

Simplify each function. List any restrictions on the domain. $$ f(x)=\frac{x^{2}-1}{x^{2}+5 x-6} $$

Simplify each complex fraction. $$ \frac{2+\frac{4}{y-7}}{\frac{4}{y-7}} $$

Solve each formula for the specified variable. \(P+\frac{a}{V^{2}}=\frac{R T}{V-b}\) for \(b\) (from physics)

Express each variation model in words. In each equation, \(k\) is the constant of variation. $$ \text { 52. } R=\frac{k L}{d^{2}} $$