A Rational Expression or Algebraic Fraction is a quotient of two polynomials.

1. A Rational Expression or Algebraic Fraction is a quotient of two polynomials. Whenever an algebraic fraction has a single term numerator and denominator, it can be simplified using the following property: where b  0, and k  0. If the denominator is equal to 0, the rational expression is undefined. This property is often called the Fundamental Principle of Fractions. We use this property to reduce fractions to the lowest terms. If reducing rational expressions with the same variable factor in the numerator and denominator, divide out pairs of identical factors, one from the numerator and one from the denominator. Each factor of the pair is lined out and converted to an understood “1”. Examples: , We can now state the Quotient Rule for Exponents. If b is any nonzero real number, and m and n are positive integers, then If the bases are the same, subtract the smaller exponent from the larger exponent. Keep the variable where the exponent is larger.

2. 2 3 Your Turn Problem #1 2. Use the quotient rule. If there is a like factor in both numerator and denominator, subtract the exponents, and keep the factor where the exponent is larger. Which has more x’s? Answer: The denominator. How many more? Answer: 2. Which has more y’s? Answer: The numerator. How many more? 6.

3. If a numerator and/or denominator contain more than one term, factor each if possible. Then divide out like factors. Solution: Your Turn Problem #2

4. The major concept of this section is to reduce fractions. The following problems will be similar to the previous examples. Factor the numerator and denominator, then divide out like factors. Solution: Your Turn Problem #3 Factor both numerator and denominator. Second, divide out like factors

5. Solution: Answer: Your Turn Problem #4 First, factor the numerator and the denominator Second, divide out like factors

6. Solution: 2 Answer: Your Turn Problem #5 1st, factor out the GCF; greatest common factor. 2nd, factor numerator and denominator. The numerator is the sum of two cubes. The denominator is the difference of two squares. 3rd, reduce any like terms.

7. Solution: Answer: Your Turn Problem #6 First, factor both numerator and denominator by grouping. Next, reduce and divide out like factors.

8. Recall the first steps of factoring: • write the terms in descending or alphabetical order • factor out a “–1” along with the GCF if the first term is negative. Example: If we have b – a, we would want to write it as –a + b. Then, factoring out a “–1”, we obtain –1(a – b). Applying these steps to the following expression: Shortcut Note: Any rational expression of the form: Solution: Your Turn Problem #7 Answer: First, rearrange into descending order Second, factor out GCF. Also, factor out the negative in the denominator. Third, divide out the like factors

9. Solution: Answer: Your Turn Problem #8 To begin factoring, rewrite the denominator in descending order. Then factor out the –1 in the denominator so the first term of the trinomial is positive. Now we can factor both numerator and denominator. Divide out like factors. The End. B.R. 12-26-06