### Key Concepts

### 8.1 Simplify Rational Expressions

**Determine the Values for Which a Rational Expression is Undefined**- Step 1. Set the denominator equal to zero.
- Step 2. Solve the equation, if possible.

**Simplified Rational Expression**- A rational expression is considered simplified if there are no common factors in its numerator and denominator.

**Simplify a Rational Expression**- Step 1. Factor the numerator and denominator completely.
- Step 2. Simplify by dividing out common factors.

**Opposites in a Rational Expression**- The opposite of $a-b$ is $b-a$.

$\begin{array}{cccccc}\frac{a-b}{b-a}=\mathrm{-1}\hfill & & & & & a\ne 0,b\ne 0,\text{a}\ne \text{b}\hfill \end{array}$

- The opposite of $a-b$ is $b-a$.

### 8.2 Multiply and Divide Rational Expressions

**Multiplication of Rational Expressions**- If $p,q,r,s$ are polynomials where $q\ne 0,s\ne 0$, then $\frac{p}{q}\xb7\frac{r}{s}=\frac{pr}{qs}$.
- To multiply rational expressions, multiply the numerators and multiply the denominators

**Multiply a Rational Expression**- Step 1. Factor each numerator and denominator completely.
- Step 2. Multiply the numerators and denominators.
- Step 3. Simplify by dividing out common factors.

**Division of Rational Expressions**- If $p,q,r,s$ are polynomials where $q\ne 0,r\ne 0,s\ne 0$, then $\frac{p}{q}\xf7\frac{r}{s}=\frac{p}{q}\xb7\frac{s}{r}$.
- To divide rational expressions multiply the first fraction by the reciprocal of the second.

**Divide Rational Expressions**- Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
- Step 2. Factor the numerators and denominators completely.
- Step 3. Multiply the numerators and denominators together.
- Step 4. Simplify by dividing out common factors.

### 8.3 Add and Subtract Rational Expressions with a Common Denominator

**Rational Expression Addition**- If $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$, then

$$\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}$$ - To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.

- If $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$, then
**Rational Expression Subtraction**- If $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$, then

$$\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}$$ - To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

- If $p,q,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}r$ are polynomials where $r\ne 0$, then

### 8.4 Add and Subtract Rational Expressions with Unlike Denominators

**Find the Least Common Denominator of Rational Expressions**- Step 1. Factor each expression completely.
- Step 2. List the factors of each expression. Match factors vertically when possible.
- Step 3. Bring down the columns.
- Step 4. Multiply the factors.

**Add or Subtract Rational Expressions**- Step 1. Determine if the expressions have a common denominator.

**Yes**– go to step 2.

**No**– Rewrite each rational expression with the LCD.- Find the LCD.
- Rewrite each rational expression as an equivalent rational expression with the LCD.

- Step 2. Add or subtract the rational expressions.
- Step 3. Simplify, if possible.

- Step 1. Determine if the expressions have a common denominator.

### 8.5 Simplify Complex Rational Expressions

**To Simplify a Rational Expression by Writing it as Division**- Step 1. Simplify the numerator and denominator.
- Step 2. Rewrite the complex rational expression as a division problem.
- Step 3. Divide the expressions.

**To Simplify a Complex Rational Expression by Using the LCD**- Step 1. Find the LCD of all fractions in the complex rational expression.
- Step 2. Multiply the numerator and denominator by the LCD.
- Step 3. Simplify the expression.

### 8.6 Solve Rational Equations

**Strategy to Solve Equations with Rational Expressions**- Step 1. Note any value of the variable that would make any denominator zero.
- Step 2. Find the least common denominator of
*all*denominators in the equation. - Step 3. Clear the fractions by multiplying both sides of the equation by the LCD.
- Step 4. Solve the resulting equation.
- Step 5. Check.

- If any values found in Step 1 are algebraic solutions, discard them.
- Check any remaining solutions in the original equation.

### 8.7 Solve Proportion and Similar Figure Applications

**Property of Similar Triangles**- If $\text{\Delta}ABC$ is similar to $\text{\Delta}XYZ$, then their corresponding angle measures are equal and their corresponding sides are in the same ratio.

**Problem Solving Strategy for Geometry Applications**- Step 1.
**Read**the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information. - Step 2.
**Identify**what we are looking for. - Step 3.
**Name**what we are looking for by choosing a variable to represent it. - Step 4.
**Translate**into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. - Step 5.
**Solve the equation**using good algebra techniques. - Step 6.
**Check**the answer in the problem and make sure it makes sense. - Step 7.
**Answer**the question with a complete sentence.

- Step 1.