1. UNIT: RATIONAL FUNCTIONS 9-4: RATIONAL EXPRESSIONS DAY 1 (SIMPLIFICATION) Essential Question: How do you simplify, multiply and divide rational expressions?

2. 9-4: Rational Expressions  A rational expression is in simplest form when its numerator and denominator are polynomials which have no common divisors.  Examples in simplest form:   Not in simplest form: 

3. 9-4: Rational Expressions  To simplify a rational expression:  Remove any GCFs that may exist  Factor if possible  Cancel common factors (parenthesis must match)  To find restrictions:  Before canceling out anything, check the denominators  Set any denominator pieces equal to 0  These give you numbers your variable cannot be

4. 9-4: Rational Expressions  Example  Simplify. State any restrictions on the variables.  Factor x 2 + 10x + 25  Factor x 2 + 9x + 20  Restrictions:  (x + 5)(x + 5) (x + 4)(x + 5) x ≠ -4, x ≠ -5

5. 9-4: Rational Expressions  Your Turn #1 (Already simplified, just reduce)  Simplify. State any restrictions on the variables.  Restrictions:  Simplified: x ≠ 0, y ≠ 0

6. 9-4: Rational Expressions  Your Turn #2 (GCF & Factor)  Simplify. State any restrictions on the variables.  Simplify -6 – 3x  Factor x 2 – 6x + 8  Restrictions: x ≠ 2, x ≠ 4 -3(x + 2) (x – 2)(x – 4)

7. 9-4: Rational Expressions  Your Turn #3 (Factor top & bottom)  Simplify. State any restrictions on the variables.  Factor 2x 2 – 3x – 2  Factor x 2 – 5x + 6  Restrictions:  (2x + 1)(x – 2) (x – 2)(x – 3) x ≠ 2, x ≠ 3

8. 9-4: Rational Expressions  Assignment  Page 511  Problems 1 – 6, all  You must show your work

9. UNIT: RATIONAL FUNCTIONS 9-4: RATIONAL EXPRESSIONS DAY 2 (MULTIPLICATION) Essential Question: How do you simplify, multiply and divide rational expressions?

10. 9-4: Rational Expressions  To multiply rational expressions:  Remove any GCFs that may exist  Factor if possible  Cancel common factors (parenthesis must match)  Multiply numerators with numerators, denominators with denominators  To find restrictions:  Before canceling out anything, check all denominators  Set any denominator pieces equal to 0  These give you numbers your variable cannot be

11. 9-4: Rational Expressions  Example  Multiply and. State any restrictions on the variables.  Factor 2x 2 + 7x + 3  Factor x 2 – 16  Factor x 2 + 8x + 15  Simplified:  Restrictions: (2x + 1)(x + 3) (x – 4)(x + 4) x ≠ 4, x ≠ -5, x ≠ -3 (x + 5)(x + 3)

12. 9-4: Rational Expressions  Y OUR T URN  Multiply and. State any restrictions on the variables.  Factor x 2 – 4  Factor x 2 – 1  Factor x 2 + 2x  Simplified:  Restrictions: (x + 2)(x – 2) (x + 1)(x – 1) x ≠ -1, x ≠ 1, x ≠ 0, x ≠ -2 x(x + 2)

13. UNIT: RATIONAL FUNCTIONS 9-4: RATIONAL EXPRESSIONS DAY 3 (DIVISION) Essential Question: How do you simplify, multiply and divide rational expressions?

14. 9-4: Rational Expressions  To divide rational expressions:  Remove any GCFs that may exist  Factor if possible  Flip division sign to multiplication, flip fraction after division sign  Cancel common factors (parenthesis must match)  Multiply numerators with numerators, denominators with denominators  To find restrictions:  Before canceling out anything, check all denominators  Set any denominator pieces equal to 0  These give you numbers your variable cannot be  Restrictions must be checked both before and after the flip

15. 9-4: Rational Expressions  Example  Divide by. State any restrictions on the variables.  Fix 4 – x (always want x to come first and be positive)  Flip division into multiplication  Simplified:  Restrictions: -1(x – 4) x ≠ - 2 / 3, x ≠ 2, y ≠ 5 / 7, x ≠ 4

16. 9-4: Rational Expressions  Y OUR T URN  Divide by. State any restrictions on the variables.  Factor x 2 +2x – 15  Factor x 2 – 16  Factor 3x – 12  Flip division into multiplication:  Restrictions:Simplified: (x + 5)(x – 3) (x + 4)(x – 4) 3(x – 4) x ≠ 4, x ≠ -4, x ≠ -1

17. 9-4: Rational Expressions  Assignment  Page 511  Problems 7 – 18, all  You must show your work