8.2 Adding and Subtracting Rational Expressions Goal 1 Determine the LCM of polynomials Goal 2 Add and Subtract Rational Expressions

LCD is 12. Find equivalent fractions using the LCD. Collect the numerators, keeping the LCD. Adding Fractions - A Review

What is the Least Common Multiple? Least Common Multiple (LCM) - smallest number or polynomial into which each of the numbers or polynomials will divide evenly. Fractions require you to find the Least Common Multiple (LCM) in order to add and subtract them! The Least Common Denominator is the LCM of the denominators.

Find the LCM of each set of Polynomials 1) 12y 2, 6x 2 LCM = 12x 2 y 2 2) 16ab 3, 5a 2 b 2, 20acLCM = 80a 2 b 3 c 3) x 2 – 2x, x 2 - 4LCM = x(x + 2)(x – 2) 4) x 2 – x – 20, x 2 + 6x + 8 LCM = (x + 4) (x – 5) (x + 2)

Remember: When adding or subtracting fractions, you need a common denominator! When Multiplying or Dividing Fractions, you don’t need a common Denominator

1. Factor, if necessary. 2. Cancel common factors, if possible. 3. Look at the denominator. 4. Reduce, if possible. 5. Leave the denominators in factored form. Steps for Adding and Subtracting Rational Expressions: If the denominators are the same, add or subtract the numerators and place the result over the common denominator. If the denominators are different, find the LCD. Change the expressions according to the LCD and add or subtract numerators. Place the result over the common denominator.

Addition and Subtraction Is the denominator the same?? Example: Simplify Simplify... Find the LCD: 6x Now, rewrite the expression using the LCD of 6x Add the fractions... = 19 6x

LCD = 15m 2 n 2 m ≠ 0 n ≠ 0 6(3mn 2 ) + 8(5n)- 7(15m) Multiply by 3mn 2 Multiply by 5n Multiply by 15m Example 1 Simplify:

LCD = 3ab a ≠ 0 b ≠ 0 Example 5 (a)(a)(b)(b)(4a) - (2b) Simplify:

Adding and Subtracting with polynomials as denominators Simplify: Simplify... Find the LCD: Rewrite the expression using the LCD of (x + 2)(x – 2) – 5x – 22 (x + 2)(x – 2)

LCD = (x + 3)(x + 1) x ≠ -1, (x + 1) (x + 3) Multiply by (x + 1) Multiply by (x + 3) Adding and Subtracting with Binomial Denominators

** Needs a common denominator 1 st ! Sometimes it helps to factor the denominators to make it easier to find your LCD. LCD: 3x 3 (2x+1) Example 6 Simplify:

LCD: (x+3) 2 (x-3) Example 7 Simplify:

x ≠ 1, -2 2x2x (x + 2)- 3x(x - 1) Example 8 Simplify:

(x + 3)(x + 2)(x + 3)(x - 1) LCD (x + 3)(x + 2)(x - 1) 3x3x x ≠ -3, -2, 1 - 2x (x - 1) (x + 2) Simplify: Example 9

(x - 3)(x - 2) (x - 2)(x - 2) 4x4x+ 5x x ≠ 3, 2 (x - 2) (x - 3) LCD (x - 3)(x - 2)(x - 2) Example 10 Simplify:

(x - 1)(x + 1)(x - 2)(x - 1) (x + 3)- (x - 4) x ≠ 1, -1, 2 (x - 2)(x + 1) LCD (x - 1)(x + 1)(x - 2) Simplify: Example 11

Pg 173 # 1 – 21 odd