8.1 Simplify Rational Expressions

Determine the Values for Which a Rational Expression is Undefined
1. Step 1. Set the denominator equal to zero.
2. 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
1. Step 1. Factor the numerator and denominator completely.
2. Step 2. Simplify by dividing out common factors.
Opposites in a Rational Expression
- The opposite of $a - b$ is $b - a$ .
  $\begin{matrix} \frac{a - b}{b - a} = −1 & a \neq 0, b \neq 0, a \neq b \end{matrix}$

8.2 Multiply and Divide Rational Expressions

Multiplication of Rational Expressions
- If $p, q, r, s$ are polynomials where $q \neq 0, s \neq 0$ , then $\frac{p}{q} \cdot \frac{r}{s} = \frac{p r}{q s}$ .
- To multiply rational expressions, multiply the numerators and multiply the denominators
Multiply a Rational Expression
1. Step 1. Factor each numerator and denominator completely.
2. Step 2. Multiply the numerators and denominators.
3. Step 3. Simplify by dividing out common factors.
Division of Rational Expressions
- If $p, q, r, s$ are polynomials where $q \neq 0, r \neq 0, s \neq 0$ , then $\frac{p}{q} \div \frac{r}{s} = \frac{p}{q} \cdot \frac{s}{r}$ .
- To divide rational expressions multiply the first fraction by the reciprocal of the second.
Divide Rational Expressions
1. Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
2. Step 2. Factor the numerators and denominators completely.
3. Step 3. Multiply the numerators and denominators together.
4. Step 4. Simplify by dividing out common factors.

Rational Expression Addition
- If $p, q, and r$ are polynomials where $r \neq 0$ , 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.
Rational Expression Subtraction
- If $p, q, and r$ are polynomials where $r \neq 0$ , 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.

Find the Least Common Denominator of Rational Expressions
1. Step 1. Factor each expression completely.
2. Step 2. List the factors of each expression. Match factors vertically when possible.
3. Step 3. Bring down the columns.
4. Step 4. Multiply the factors.
Add or Subtract Rational Expressions
1. 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.
2. Step 2. Add or subtract the rational expressions.
3. Step 3. Simplify, if possible.

To Simplify a Rational Expression by Writing it as Division
1. Step 1. Simplify the numerator and denominator.
2. Step 2. Rewrite the complex rational expression as a division problem.
3. Step 3. Divide the expressions.
To Simplify a Complex Rational Expression by Using the LCD
1. Step 1. Find the LCD of all fractions in the complex rational expression.
2. Step 2. Multiply the numerator and denominator by the LCD.
3. Step 3. Simplify the expression.

Strategy to Solve Equations with Rational Expressions
1. Step 1. Note any value of the variable that would make any denominator zero.
2. Step 2. Find the least common denominator of all denominators in the equation.
3. Step 3. Clear the fractions by multiplying both sides of the equation by the LCD.
4. Step 4. Solve the resulting equation.
5. Step 5. Check.
- If any values found in Step 1 are algebraic solutions, discard them.
- Check any remaining solutions in the original equation.

Property of Similar Triangles
- If $Δ A B C$ is similar to $Δ X Y Z$ , then their corresponding angle measures are equal and their corresponding sides are in the same ratio.
Problem Solving Strategy for Geometry Applications
1. 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.
2. Step 2. Identify what we are looking for.
3. Step 3. Name what we are looking for by choosing a variable to represent it.
4. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Step 5. Solve the equation using good algebra techniques.
6. Step 6. Check the answer in the problem and make sure it makes sense.
7. Step 7. Answer the question with a complete sentence.