Chapter 6

Solve equation. If a solution is extraneous, so indicate. \(x^{-1}-3=4 x^{-1}\) \(\left(\text {Hint: Use } x^{-n}=\frac{1}{x^{n}}\right)\)

What is a complex fraction?

Perform the operations and simplify the result when possible. $$\frac{x-y}{2}+\frac{x+y}{3}$$

Perform each division. \(\frac{x^{5}+3 x+2}{x^{3}+1+2 x}\)

Perform the operations and simplify. $$ \frac{p^{3}-q^{3}}{p^{2}-q^{2}} \cdot \frac{q^{2}+p q}{p^{3}+p^{2} q+p q^{2}} $$

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

Use synthetic division to perform each division. $$ \frac{-5 x^{5}+4 x^{4}+30 x^{3}+2 x^{2}+20 x+3}{x-3} $$

Solve equation. If a solution is extraneous, so indicate. \(x^{-1}+2=3 x^{-1}\) \(\left(\text {Hint: Use } x^{-n}=\frac{1}{x^{n}}\right)\)

Perform the operations and simplify the result when possible. $$\frac{a+b}{3}+\frac{a-b}{7}$$

Drafting. In a scale drawing, a 280 -foot antenna tower is drawn \(7 \frac{1}{2}\) inches high. The building next to it is drawn \(2 \frac{1}{4}\) inches high. How tall is the actual building?

Perform each division. \(\frac{9 a^{4}+6 a^{3}+55 a^{2}+18 a+81}{3 a^{2}+a+9}\)

Perform the operations and simplify. $$ \frac{x^{2}-4}{2 b-b x} \div \frac{x^{2}+4 x+4}{2 b+b x} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{a^{3}-a b^{2}}{a b^{2}-4 a^{2} b+3 a^{3}} $$

Use synthetic division to perform each division. $$ \frac{9 a^{3}+3 a^{2}-21 a-7}{a+\frac{1}{3}} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{5}{y-1}+\frac{3}{y-3}=\frac{8}{y-2}\)

What does it mean when we say that $$ \frac{\frac{1}{x+2}}{1+\frac{1}{x+2}} \quad \text { and } \quad \frac{1}{x+3} $$ are equivalent expressions?

Perform the operations and simplify the result when possible. $$\frac{3 x^{2}+3 x}{x^{2}-5 x+6}-\frac{3 x^{2}-3 x+12}{x^{2}-5 x+6}$$

Perform each division. Divide \(11 x^{2}-4 x+8 x^{4}-6 x^{3}+3\) by \(3+4 x^{2}-x\)

Perform the operations and simplify. $$ \frac{10 r^{2} s}{6 r s^{2}} \cdot \frac{3 r^{3}}{2 r s} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{s x+4 s-3 x-12}{s x+4 s+6 x+24} $$

Use synthetic division to perform each division. $$ \frac{8 t^{3}-4 t^{2}+2 t-1}{t-\frac{1}{2}} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{3+2 a}{a^{2}+6+5 a}+\frac{2-5 a}{a^{2}-4}=\frac{2-3 a}{a^{2}-6+a}\)

Perform the operations and simplify the result when possible. $$\frac{2 m^{2}-7}{m^{4}-9}+\frac{4-m^{2}}{m^{4}-9}$$

Use similar triangles to solve each problem. Washington, D.C. The Washington Monument casts a shadow of \(166 \frac{1}{2}\) feet at the same time as a 5 -foot-tall tourist casts a shadow of \(1 \frac{1}{2}\) feet. Find the height of the monument.

Perform each division. Divide \(4 x-2+x^{4}-x^{2}-x^{3}\) by \(x-1+x^{2}\)

Perform the operations and simplify. $$ \frac{3 a^{3} b}{25 c d^{3}} \cdot \frac{5 c d^{2}}{6 a b} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{a x+b y+a y+b x}{a+b} $$

Use synthetic division to perform each division. $$ \frac{4 x^{4}+12 x^{3}-x^{2}-x+12}{x+3} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{3}{s-2}+\frac{s-14}{2 s^{2}-3 s-2}-\frac{4}{2 s+1}=0\)

Solve each equation. $$ \frac{8(a-5)}{3}=2(a-4) $$

Perform the operations and simplify the result when possible. $$\frac{a^{2}+a b}{a^{3}-b^{3}}-\frac{b^{2}}{b^{3}-a^{3}}$$

Use similar triangles to solve each problem. Towers. \(\quad\) A cell phone tower casts a shadow of 75 feet at the same time that a 6 -foot-tall tree has a shadow that is 3 feet 9 inches long. How tall is the cell phone tower?

Perform each division. \(\frac{15 x^{2}+9 x-3}{27}\)

Perform the operations and simplify. $$ 10(h-9) \frac{h-3}{9-h} $$

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

Use synthetic division to perform each division. $$ \frac{x^{4}-9 x^{3}+x^{2}-7 x-20}{x-9} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{1}{y^{2}-2 y-3}+\frac{1}{y^{2}-4 y+3}-\frac{1}{y^{2}-1}=0\)

Solve each equation. $$ \frac{3 t^{2}}{5}+\frac{7 t}{10}=\frac{3 t+6}{5} $$

Perform the operations and simplify the result when possible. $$\frac{y^{2}-3 x y}{x^{3}-y^{3}}-\frac{x^{2}+4 x y}{y^{3}-x^{3}}$$

Perform each division. \(\frac{8 x^{2}+12 x+9}{6}\)

Perform the operations and simplify. $$ r(r-25) \frac{r+4}{r-25} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{6 x^{2}-7 x-5}{2 x^{2}+5 x+2} $$

Use synthetic division to perform each division. $$ \frac{3 x^{3}-25 x^{2}+10 x-16}{x-8} $$

Solve equation. If a solution is extraneous, so indicate. \(\frac{x}{x+2}=1-\frac{3 x+2}{x^{2}+4 x+4}\)

Solve each equation. $$ a^{4}-13 a^{2}+36=0 $$

Perform the operations and simplify the result when possible. $$2 x+3+\frac{1}{x+1}$$

Use similar triangles to solve each problem. Flight Paths. \(\quad\) An airplane ascends 150 feet as it flies a horizontal distance of \(1,000\) feet. How much altitude will it gain as it flies a horizontal distance of 1 mile? (Hint: \(5,280 \text { feet }=1 \text { mile. })\)

Perform the operations and simplify. $$ \frac{2 x^{2}+5 x y+3 y^{2}}{3 x^{2}-5 x y+2 y^{2}} \div \frac{2 x^{2}+x y-3 y^{2}}{3 x^{2}-5 x y+2 y^{2}} $$

Simplify each expression. If an expression cannot be simplified, write "Does not simplify." $$ \frac{3 x+6 y}{x+2 y} $$

Use synthetic division to perform each division. $$ \frac{2 x^{3}+3 x^{2}-8 x+3}{x+3} $$