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

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

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

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

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.

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.

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

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

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.

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

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

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

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

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

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

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

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

Example: X Factoring Simplify the rational expression. (c + 2)(c – 3)

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

Example: Simplify the rational expression.

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

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

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

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: (1– c)