Outro W10

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.
    $\frac{a-b}{b-a}=-1$    a≠0, b≠0, a≠b

8.2 Multiply and Divide Rational Expressions

• Multiplication of Rational Expressions
  • If p, q, r, s are polynomials where q≠0, s≠0 , then $\frac{p}{q}\cdot\frac{r}{s}=\frac{pr}{qs}$
  • 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≠0, r≠0, s≠0  then $\frac{p}{q}\div\frac{r}{s}=\frac{p}{q}\cdot\frac{r}{s}$.
  • 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.

8.3 Add and Subtract Rational Expressions with a Common Denominator

• Rational Expression Addition
  • If p, q, and r are polynomials where r≠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≠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.

8.4 Add and Subtract Rational Expressions with Unlike Denominators

• 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.