1. Module: 0 Part 4: Rational Expressions
Obj: Simplify rational expressions, complex fractions
Evi: Students will solve rational expressions

2. Rational Expressions
The set of real numbers for which an algebraic expression is defined is the domain of the expression.
𝑥+2 𝑥−3
This is a rational expression the domain is all real numbers expect x=3. If x=3, the bottom is zero and then Channing Tatum dies… we don’t want that.

3. Simplifying a Rational Expression
Step 1: Factor Completely
Step 2: Divide out common factors
Step 3: State answer and restrictions
𝑥 2 +4𝑥−12 3𝑥−6
(𝑥+6)(𝑥−2) 3(𝑥−2)
𝑥 𝑥≠2

4. Another Example
12+𝑥− 𝑥 2 2 𝑥 2 −9𝑥+4
− 𝑥+3 2𝑥− 𝑥=4, ± 1 2

5. Multiplying Rational Expressions
Step 1: Factor both of them
Step 2: Cancel out common factors
Step 3: Figure out domain
2 𝑥 2 +𝑥−6 𝑥 2 +4𝑥−5 ∗ 𝑥 3 −3 𝑥 2 +2𝑥 4 𝑥 2 −6𝑥
(2+3)(𝑥+2) (𝑥+5)(𝑥−1) ∗ 𝑥( 𝑥 2 −3𝑥+2) 2𝑥(2𝑥−3)
(2𝑥−3)(𝑥+2) (𝑥+5)(𝑥−1) ∗ 𝑥(𝑥−2)(𝑥−1) 2𝑥(2𝑥−3)
(𝑥+2) (𝑥+5) ∗ (𝑥−2) 2
(𝑥+2)(𝑥−2) 2(𝑥+5) 𝑥≠0, 1, −5, 3 2

6. Dividing Rational Expressions
Same as multiplying but you have to switch so that it becomes a multiply by the reciprocal

7. Adding and Subtracting
Step 1: Get common denominator
Step 2: Simplify
𝑥 𝑥−3 − 2 3𝑥+4
3𝑥+4 𝑥− 𝑥−3 2 (𝑥−3)(3𝑥+4)
3 𝑥 2 +2𝑥+6 (𝑥−3)(3𝑥+4)

8. Another Example
3 𝑥−1 − 2 𝑥 + 𝑥+3 𝑥 2 −1
2( 𝑥 2 +3𝑥+1) 𝑥(𝑥+1)(𝑥−1)

9. Complex Fractions 2 𝑥 −3 1− 1 𝑥−1 2 𝑥 −3 2 𝑥 − 3𝑥 𝑥 2−3𝑥 𝑥
Step 1: Separate fractions
Step 2: Find Common Denominator
Step 3: Simplify
Step 4: Multiple by reciprocal
Step 5: Simplify
2 𝑥 −3 1− 1 𝑥−1
2 𝑥 − 𝑥 − 3𝑥 𝑥 −3𝑥 𝑥
1− 1 𝑥− 𝑥−1−1 𝑥− 𝑥−2 𝑥−1
2−3𝑥 𝑥 𝑥−2 𝑥− −3𝑥 𝑥 ÷ 𝑥−2 𝑥− −3𝑥 𝑥 ∗ 𝑥−1 𝑥−2
(2−3𝑥)(𝑥−1) 𝑥(𝑥−2)