8.1 Simplify Rational Expressions

If you miss a problem, go back to the section listed and review the material.

Simplify: $\frac{90 y}{15 y^{2}} .$
If you missed this problem, review the reference.

Factor: $6 x^{2} - 7 x + 2 .$
If you missed this problem, review the reference.

Factor: $n^{3} + 8 .$
If you missed this problem, review the reference.

Determine the values for which the rational expression is undefined:

ⓐ $\frac{9 y}{x}$ ⓑ $\frac{4 b - 3}{2 b + 5}$ ⓒ $\frac{x + 4}{x^{2} + 5 x + 6}$

Determine the values for which the rational expression is undefined:

ⓐ $\frac{3 y}{x}$ ⓑ $\frac{8 n - 5}{3 n + 1}$ ⓒ $\frac{a + 10}{a^{2} + 4 a + 3}$

Determine the values for which the rational expression is undefined:

ⓐ $\frac{4 p}{5 q}$ ⓑ $\frac{y - 1}{3 y + 2}$ ⓒ $\frac{m - 5}{m^{2} + m - 6}$

Evaluate $\frac{y + 1}{2 y - 3}$ for each value:

ⓐ $y = 1$ ⓑ $y = −3$ ⓒ $y = 0$

Evaluate $\frac{5 x - 1}{2 x + 1}$ for each value:

ⓐ $x = 1$ ⓑ $x = −1$ ⓒ $x = 0$

Evaluate $\frac{x^{2} + 1}{x^{2} - 3 x + 2}$ for each value:

ⓐ $x = 0$ ⓑ $x = −1$ ⓒ $x = 3$

Evaluate $\frac{x^{2} + x - 6}{x^{2} - 9}$ for each value:

ⓐ $x = 0$ ⓑ $x = −2$ ⓒ $x = 1$

Evaluate $\frac{2 a^{3} b}{a^{2} + 2 a b + b^{2}}$ for each value:

ⓐ $a = −1 , b = 2$ ⓑ $a = 0 , b = −1$ ⓒ $a = 1 , b = \frac{1}{2}$

Evaluate $\frac{a^{2} - b^{2}}{8 a b^{3}}$ for each value:

ⓐ $a = 1 , b = −1$ ⓑ $a = \frac{1}{2} , b = −1$ ⓒ $a = −2 , b = 1$

Simplify: $- \frac{45}{81} .$

Simplify: $- \frac{42}{54} .$

Simplify: $\frac{4 x^{2} y}{12 x y^{2}} .$

Simplify: $\frac{16 x^{2} y}{2 x y^{2}} .$

Simplify: $\frac{3 x - 6}{2 x - 4} .$

Simplify: $\frac{7 y + 35}{5 y + 25} .$

Simplify: $\frac{x^{2} + 5 x + 6}{x^{2} + 8 x + 12} .$

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

Simplify: $\frac{x^{2} - 3 x - 10}{x^{2} + x - 2} .$

Simplify: $\frac{y^{2} + y - 42}{y^{2} - 36} .$

Simplify: $\frac{x^{2} + x - 6}{x^{2} - 4} .$

Simplify: $\frac{x^{2} + 8 x + 7}{x^{2} - 49} .$

Simplify: $\frac{p^{3} - 2 p^{2} + 2 p - 4}{p^{2} - 7 p + 10} .$

Simplify: $\frac{y^{3} - 3 y^{2} + y - 3}{y^{2} - y - 6} .$

Simplify: $\frac{p^{3} - p^{2} + 2 p - 2}{p^{2} + 4 p - 5} .$

Simplify: $\frac{2 n^{2} - 14 n}{4 n^{2} - 16 n - 48} .$

Simplify: $\frac{2 n^{2} - 10 n}{4 n^{2} - 16 n - 20} .$

Simplify: $\frac{4 x^{2} - 16 x}{8 x^{2} - 16 x - 64} .$

Simplify: $\frac{3 b^{2} - 12 b + 12}{6 b^{2} - 24} .$

Simplify: $\frac{2 x^{2} - 12 x + 18}{3 x^{2} - 27} .$

Simplify: $\frac{5 y^{2} - 30 y + 25}{2 y^{2} - 50} .$

Simplify: $\frac{m^{3} + 8}{m^{2} - 4} .$

Simplify: $\frac{p^{3} - 64}{p^{2} - 16} .$

Simplify: $\frac{x^{3} + 8}{x^{2} - 4} .$

Simplify: $\frac{x - 8}{8 - x} .$

Simplify: $\frac{y - 2}{2 - y} .$

Simplify: $\frac{n - 9}{9 - n} .$

Simplify: $\frac{10 - 2 y}{y^{2} - 25} .$

Simplify: $\frac{3 y - 27}{81 - y^{2}} .$

Simplify: $\frac{x^{2} - 4 x - 5}{25 - x^{2}} .$

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

Determine the values for which the rational expression is undefined.

ⓐ $\frac{2 x}{z}$ ⓑ $\frac{4 p - 1}{6 p - 5}$ ⓒ $\frac{n - 3}{n^{2} + 2 n - 8}$

Determine the values for which the rational expression is undefined.

ⓐ $\frac{4 x^{2} y}{3 y}$ ⓑ $\frac{3 x - 2}{2 x + 1}$ ⓒ $\frac{u - 1}{u^{2} - 3 u - 28}$

Evaluate the rational expression for the given values.

$\frac{2 x}{x - 1}$

ⓐ $x = 0$ ⓑ $x = 2$ ⓒ $x = −1$

Evaluate the rational expression for the given values.

$\frac{2 p + 3}{p^{2} + 1}$

ⓐ $p = 0$ ⓑ $p = 1$ ⓒ $p = −2$

Evaluate the rational expression for the given values.

$\frac{y^{2} + 5 y + 6}{y^{2} - 1}$

ⓐ $y = 0$ ⓑ $y = 2$ ⓒ $y = −2$

Evaluate the rational expression for the given values.

$\frac{a^{2} - 4}{a^{2} + 5 a + 4}$

ⓐ $a = 0$ ⓑ $a = 1$ ⓒ $a = −2$

Evaluate the rational expression for the given values.

$\frac{x^{2} + 3 x y + 2 y^{2}}{2 x^{3} y}$

1. ⓐ $x = 1 , y = −1$
2. ⓑ $x = 2 , y = 1$
3. ⓒ $x = −1 , y = −2$

Evaluate the rational expression for the given values.

$\frac{m^{2} - 4 n^{2}}{5 m n^{3}}$

1. ⓐ $m = 2 , n = 1$
2. ⓑ $m = −1 , n = −1$
3. ⓒ $m = 3 , n = 2$

Simplify.

$- \frac{4}{52}$

Simplify.

$\frac{56}{63}$

Simplify.

$\frac{6 a b^{2}}{12 a^{2} b}$

Simplify.

$\frac{8 m^{3} n}{12 m n^{2}}$

Simplify.

$\frac{3 a + 6}{4 a + 8}$

Simplify.

$\frac{3 c - 9}{5 c - 15}$

Simplify.

$\frac{7 m + 63}{5 m + 45}$

Simplify.

$\frac{12 p - 240}{5 p - 100}$

Simplify.

$\frac{a^{2} - a - 12}{a^{2} - 8 a + 16}$

Simplify.

$\frac{y^{2} + 3 y - 4}{y^{2} - 6 y + 5}$

Simplify.

$\frac{x^{2} - 25}{x^{2} + 2 x - 15}$

Simplify.

$\frac{y^{2} - 2 y - 3}{y^{2} - 9}$

Simplify.

$\frac{y^{3} + y^{2} + y + 1}{y^{2} + 2 y + 1}$

Simplify.

$\frac{x^{3} - 2 x^{2} - 25 x + 50}{x^{2} - 25}$

Simplify.

$\frac{3 a^{2} + 15 a}{6 a^{2} + 6 a - 36}$

Simplify.

$\frac{−5 c^{2} - 10 c}{−10 c^{2} + 30 c + 100}$

Simplify.

$\frac{3 m^{2} + 30 m + 75}{4 m^{2} - 100}$

Simplify.

$\frac{5 r^{2} + 30 r - 35}{r^{2} - 49}$

Simplify.

$\frac{t^{3} - 27}{t^{2} - 9}$

Simplify.

$\frac{w^{3} + 216}{w^{2} - 36}$

Simplify each rational expression.

$\frac{a - 5}{5 - a}$

Simplify each rational expression.

$\frac{11 - c}{c - 11}$

Simplify each rational expression.

$\frac{12 - 2 x}{x^{2} - 36}$

Simplify each rational expression.

$\frac{4 v - 32}{64 - v^{2}}$

Simplify each rational expression.

$\frac{y^{2} - 11 y + 24}{9 - y^{2}}$

Simplify each rational expression.

$\frac{a^{2} - 5 a - 36}{81 - a^{2}}$

Tax Rates For the tax year 2015, the amount of tax owed by a single person earning between $37,450 and $90,750, can be found by evaluating the formula $0.25 x - 4206.25 ,$ where x is income. The average tax rate for this income can be found by evaluating the formula $\frac{0.25 x - 4206.25}{x} .$ What would be the average tax rate for a single person earning $50,000?

Explain how you find the values of x for which the rational expression $\frac{x^{2} - x - 20}{x^{2} - 4}$ is undefined.