1. Operations on Rational Expressions Digital Lesson

2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2 Rational Expressions Rational expressions are fractions in which the numerator and denominator are polynomials and the denominator does not equal zero. Example: Simplify., x – 3  0, x  3

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 Multiplying Rational Expression 1. Factor the numerator and denominator of each fraction. 2. Multiply the numerators and denominators of each fraction. 4. Write the answer in simplest form. 3. Divide by the common factors. To multiply rational expressions:

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Example: Multiplication Factor the numerator and denominator of each fraction. Divide by the common factors. Write the answer in simplest form. Multiply.Example: Multiply.

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 Dividing Rational Expressions 1. Multiply the dividend by the reciprocal of the divisor. The reciprocal of is. 2. Multiply the numerators. Then multiply the denominators. 4. Write the answer in simplest form. 3. Divide by the common factors. To divide rational expressions:

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Example: Division Example: Divide. Multiply by the reciprocal of the divisor. Factor and multiply. Divide by the common factors. Simplest form

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Least Common Multiple The least common multiple (LCM) of two or more numbers is the least number that contains the prime factorization of each number. Examples: 1. Find the LCM of 10 and 4. 2. Find the LCM of 4x 2 + 4x and x 2 + 2x + 1. 4x 2 + 4x = (4x)(x +1) = 2 2 x (x + 1) x 2 + 2x + 1 = (x +1)(x +1) LCM = 2 2 x (x +1)(x +1) factors of 4x 2 + 4x factors of x 2 + 2x + 1 10 = (5 2) LCM = 2 2 5 factors of 10 factors of 4 4 = (2 2) = 4x 3 + 8x 2 + 4x = 20

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 LCM Least Common Multiple of Denominator s Fractions can be expressed in terms of the least common multiple of their denominators. Example: Write the fractions and in terms of the LCM of the denominators. The LCM of the denominators is 12x 2 (x – 2).

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 Adding and Subtracting Rational Expressions 1.If necessary, rewrite the fractions with a common denominator. To add rational expressions: To subtract rational expressions: 2. Add the numerators of each fraction. 1.If necessary, rewrite the fractions with a common denominator. 2. Subtract the numerators of each fraction.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 Examples: Addition & Subtraction Example: Add. Example: Subtract.

11. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Different Denominators Two rational expressions with different denominators can be added or subtracted after they are rewritten with a common denominator. Example: Add.

12. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 Example: Subtract Example: Subtract. Add numerators. Factor. Divide. Simplest form