1. Warm-up Factor each polynomial. 1. 4x 2 – 6x2. 15y 3 + 20y 3. n 2 + 4n + 34. p 2 – p – 42

2. Warm-up Factor each polynomial. 1. 4x 2 – 6x2. 15y 3 + 20y Common Factor: 2x 2x(2x– 3) Common Factor: 5y 5y(3y 2 + 4)

3. Warm-up Factor each polynomial. 3. n 2 + 4n + 34. p 2 – p – 42 3 4 13 (n + 1)(n + 3) -42 6-7 (p + 6)(p – 7)

4. Section 11.1 Simplifying Rational Expressions Standards: 12.0 Objective: I will simplify a rational expression.

5. Rational Expression: A fraction with polynomials as the numerator and/or denominator. The value of the denominator cannot be 0 because division by zero is undefined.

6. Simplifying: ① Factor the numerator, if possible. ② Factor the denominator, if possible. ③ Divide (cross out) the common factors. ④ Multiply any remaining factors. You cannot cross out sums, only products.

7. Can It Be Simplified? no, the x in the denominator is not a product yes, (y + 2) is a product no, (a + b) is not the same as (a – b) and there are no products

8. Remember: When all of the factors cancel out, use 1, not 0. 1 is a factor of every number.

9. Factoring: Use the X method for x 2 + bx + c. Use the BOX method for ax 2 + bx + c. Remember, factored form is (__x +/- __)(__x +/- __). Factor by finding a common factor. Factor using prime factorization.

10. Example: Prime Factorization Simplify the rational expression. 2  7  c

11. Example: Common Factor Simplify the rational expression. 6x + 18 Common Factor: 6 6( 66 x+ 3) = 6

12. Example: Common Factor Simplify the rational expression. 4t + 20 Common Factor: 4 4( 44 t+ 5)

13. Example: Common Factor Simplify the rational expression. 8m − 4 Common Factor: 4 4( 44 2m− 1) = 4

14. Example: Simplify the rational expression. 3x + 9 Common Factor: 3 3( 33 x+ 3)

15. 6 -7 -6 Example: X Factoring Simplify the rational expression. 2x − 2 Common Factor: 2 2( 22 x− 1)

16. -20 4-5 Example: X Factoring Simplify the rational expression. 3x + 12 Common Factor: 3 3( 33 x+ 4)

17. 3 -4 -3 Example: X Factoring Simplify the rational expression. 2a − 2 Common Factor: 2 2( 22 a− 1)

18. -6 2-3 6 5 23 Example: X Factoring Simplify the rational expression. (c + 2)(c – 3)

19. Recognizing Opposites: Remember, you can only cancel out binomials if they are exactly alike. If the signs are opposite, then factor out a -1.

20. Example: Simplify the rational expression.

21. Example: Simplify the rational expression. (m + 8)(m – 8) = m = 8

22. Example: Simplify the rational expression. 5x – 15 Common Factor: 5 5( 55 x– 3) (3 + x)(3 – x) = x = 3

23. Example: Simplify the rational expression. 8 – 4r Common Factor: 4 4( 44 2– r) -8 4 -24 (r – 2)(r + 4)

24. Example: Simplify the rational expression. 2c 2 – 2 Common Factor: 2 2( 22 c2c2 – 1) 2(c + 1)(c – 1) = 1= c 3 – 3c Common Factor: 3 33 3(1– c)