Adding and Subtracting Rational Expressions

1. Adding and Subtracting Rational Expressions Essential Questions • How do we add and subtract rational expressions? • How do we simplify complex fractions? Holt McDougal Algebra 2 Holt Algebra 2

2. Adding and subtracting rational expressions is similar to adding and subtracting fractions. To add or subtract rational expressions with like denominators, add or subtract the numerators and use the same denominator.

3. + x – 3 x – 2 x + 4 x + 4 Example 1: Adding and Subtracting Rational Expressions with Like Denominators Add or subtract. Identify any x-values for which the expression is undefined. Add the numerators. Combine like terms. The expression is undefined at x = –4 because this value makes x + 4 equal 0.

4. – 3x – 4 6x + 1 x2 + 1 x2 + 1 Example 2: Adding and Subtracting Rational Expressions with Like Denominators Add or subtract. Identify any x-values for which the expression is undefined. - - Change the signs of the numerator when subtracting. + Rewrite the numerator. Combine like terms. There is no real value of x for which x2 + 1 = 0; the expression is always defined.

5. + 6x + 5 3x – 1 x2 – 3 x2 – 3 The expression is undefined at x = ± because this value makes x2 – 3 equal 0. Example 3: Adding and Subtracting Rational Expressions with Like Denominators Add or subtract. Identify any x-values for which the expression is undefined. Add the numerators. Combine like terms.

6. 1 – 3 2x2–3x –2 3x2– 5 3x – 1 3x – 1 The expression is undefined at x = because this value makes 3x – 1 equal 0. Example 4: Adding and Subtracting Rational Expressions with Like Denominators Add or subtract. Identify any x-values for which the expression is undefined. - + + Change the signs of the numerator when subtracting. + Rewrite the numerator. Combine like terms.

7. To add or subtract rational expressions with unlike denominators, first find the least common denominator (LCD). The LCD is the least common multiple of the polynomials in the denominators.

8. Example 5: Finding the Least Common Multiple of Polynomials Find the least common multiple for each pair. A. 4x2y3 and 6x4y5 4x2y3 = 2 2 x2  y3 6x4y5 = 3 2  x4  y5 2  2 3  x4 = 12x4y5. The LCM is  y5 B. x2 – 2x – 3 and x2 – x – 6 x2 – 2x – 3 = (x – 3) (x + 1) x2 – x – 6= (x – 3) (x + 2) (x – 3) (x + 2). (x + 1) The LCM is

9. Example 6: Finding the Least Common Multiple of Polynomials Find the least common multiple for each pair. A. 4x3y7 and 3x5y4 4x3y7 = 2 2 x3  y7 3x5y4 = 3 x5  y4 2  2 3  x5 = 12x5y7. The LCM is  y7 B. x2 – 4 and x2 + 5x + 6 x2 – 4 = (x – 2) (x + 2) x2 + 5x + 6= (x + 2) (x + 3) (x – 2) (x + 3). (x + 2) The LCM is

10. Lesson 6.3 Practice A