2. Addition and Subtraction with Like Denominators Let p, q, and r represent polynomials where q ≠ 0. To add or subtract when denominators are the same, add or subtract the numerators and keep the same denominator 1. 2. Addition and Subtraction with Like Denominators Let p, q, and r represent polynomials where q ≠ 0. To add or subtract when denominators are the same, add or subtract the numerators and keep the same denominator 1. 2.

3. The fractions have the same denominator. Add term in the numerators, and write the result over the common denominator. Simplify to lowest terms. Solution

4. The fractions have the same denominator. Subtract the terms in the numerators, and write the result over the common denominator. Simplify the numerator. Factor the numerator and denominator to determine if the rational expression can be simplified. Simplify to lowest terms.

5. The fractions have the same denominator. Subtract the terms in the numerators, and write the result over the common denominator. Simplify the numerator. Factor the numerator and denominator to determine if the rational expression can be simplified. Simplify to lowest terms.

6.  Factor the denominators of each rational expression.  Identify the LCD › To find the least common multiple of two or more expressions:  Find the prime factorization of each expression  Form a product that contains each factor the greatest number of times that it occurs in any one prime factorization  Rewrite each rational expression as an equivalent expression with the LCD as its denominator.  Add or subtract the numerators, and write the result over the common denominator.  Simplify to lowest terms.

7. Solution We write the prime factorizations of The factors 3, 7, and x must appear in the LCM if 21x is to be a factor of the LCM. The factors 3, x, and x must appear in the LCM if 3x² is to be a factor of the LCM

8. To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator. ADDING RATIONAL EXPRESSIONS So the common denominator needs each of these factors. This fraction needs (x + 5) This fraction needs nothing simplify distribute Reduce common factors 1 Solution

9. To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator. ADDING RATIONAL EXPRESSIONS The expressions d - 7 and 7- d are opposites and differ by a factor of - 1 Therefore, multiply the numerator and denominator of either expression by -1 to obtain a common denominator. Simplify Solution Add the terms in the numerators, and write the result over the common denominator.

10. To subtract rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator. SUBTRACTING RATIONAL EXPRESSIONS So the common denominator needs each of these factors.The LCD is 6. This fraction needs (2) This fraction needs (3) simplify distribute Reduce common factors Solution NONE

11. Subtracting rational expressions is much like adding, you must have a common denominator. The important thing to remember is that you must subtract each term of the second rational function. SUBTRACTING RATIONAL EXPRESSIONS So a common denominator needs each of these factors. This fraction needs (x + 2) This fraction needs (x + 6)) Distribute the negative to each term. FOIL

12. Solutions Factoring, Note that x ≠ -2, 2 Multiplying by -1/-1 since 2-x is the opposite of x-2

13. The LCD is (x-2)(x+2) Removing a factor equal to 1: that x ≠ -2 Multiplying by 1 to get the LCD