Factor

Objectives Add and subtract rational expressions with like denominators. Add and subtract rational expressions with unlike denominators.

The rules for adding rational expressions are the same as the rules for adding fractions. If the denominators are the same, you add the numerators and keep the common denominator.

Example 1A: Adding Rational Expressions with Like Denominators Add. Simplify your answer. Combine like terms in the numerator. Divide out common factors. Simplify.

Example 1B: Adding Rational Expressions with Like Denominators Add. Simplify your answer. Combine like terms in the numerator. Factor. Divide out common factors. Simplify.

Check It Out! Example 1a Add. Simplify your answer. Combine like terms in the numerator. Divide out common factors. = 2 Simplify.

Check It Out! Example 1b Add. Simplify your answer. Combine like terms in the numerator. Factor. Divide out common factors. Simplify.

To subtract rational expressions with like denominators, remember to add the opposite of each term in the second numerator.

Example 2: Subtracting Rational Expressions with Like Denominators Subtract. Simplify your answer. Subtract numerators. Combine like terms. Factor. Divide out common factors. Simplify.

Make sure you add the opposite of all the terms in the numerator of the second expression when subtracting rational expressions. Caution

Check It Out! Example 2a Subtract. Simplify your answer. Subtract numerators. Combine like terms. Factor. Divide out common factors. Simplify.

Check It Out! Example 2b Subtract. Simplify your answer. Subtract numerators. Combine like terms. Factor. There are no common factors.

As with fractions, rational expressions must have a common denominator before they can be added or subtracted. If they do not have a common denominator, you can use the least common multiple, or LCM, of the denominators to find one. To find the LCM, write the prime factorization of both expressions. Use each factor the greatest number of times it appears in either expression.

Example 3A: Identifying the Least Common Multiple Find the LCM of the given expressions. 12x2y, 9xy3 Write the prime factorization of each expression. Use every factor of both expressions the greatest number of times it appears in either expression. 9xy3 = 3  3  x  y  y  y 12x2y = 2  2  3  x  x  y LCM = 2  2  3  3  x  x  y  y  y = 36x2y3

Example 3B: Identifying the Least Common Multiple Find the LCM of the given expressions. c2 + 8c + 15, 3c2 + 18c + 27 Factor each expression. 3c2 + 18c + 27 = 3(c2 +6c +9) = 3(c + 3)(c + 3) Use every factor of both expressions the greatest number of times it appears in either expression. c2 + 8c + 15 = (c + 3) (c + 5) LCM = 3(c + 3)2(c + 5)

Check It Out! Example 3a Find the LCM of the given expressions. 5f2h, 15fh2 Write the prime factorization of each expression. Use every factor of both expressions the greatest number of times it appears in either expression. 5f2h = 5  f  f  h 15fh2 = 3  5  f  h  h LCM = 3  5  f  f  h  h = 15f2h2

Check It Out! Example 3b Find the LCM of the given expressions. x2 – 4x – 12, (x – 6)(x +5) Factor each expression. x2 – 4x – 12 = (x – 6) (x + 2) Use every factor of both expressions the greatest number of times it appears in either expression. (x – 6)(x +5) = (x – 6)(x + 5) LCM = (x – 6)(x + 5)(x + 2)

The LCM of the denominators of fractions or rational expressions is also called the least common denominator, or LCD. You use the same method to add or subtract rational expressions.

Example 4A: Adding and Subtracting with Unlike Denominators Add or subtract. Simplify your answer. Identify the LCD. 5n3 = 5  n  n  n Step 1 2n2 = 2  n  n LCD = 2  5  n  n  n = 10n3 Multiply each expression by an appropriate form of 1. Step 2 Step 3 Write each expression using the LCD.

Example 4A Continued Add or subtract. Simplify your answer. Step 4 Add the numerators. Step 5 Factor and divide out common factors. Step 6 Simplify.

Example 4B: Adding and Subtracting with Unlike Denominators. Add or subtract. Simplify your answer. Step 1 The denominators are opposite binomials. The LCD can be either w – 5 or 5 – w. Identify the LCD. Step 2 Multiply the first expression by to get an LCD of w – 5. Step 3 Write each expression using the LCD.

Example 4B Continued Add or Subtract. Simplify your answer. Step 4 Subtract the numerators. Step 5, 6 No factoring needed, so just simplify.

Check It Out! Example 4a Add or subtract. Simplify your answer. 3d 3  d 2d3 = 2  d  d  d LCD = 2  3 d  d  d = 6d3 Step 1 Identify the LCD. Multiply each expression by an appropriate form of 1. Step 2 Step 3 Write each expression using the LCD.

Check It Out! Example 4a Continued Add or subtract. Simplify your answer. Step 4 Subtract the numerators. Step 5 Factor and divide out common factors. Step 6 Simplify.

Check It Out! Example 4b Add or subtract. Simplify your answer. Step 1 Factor the first term. The denominator of second term is a factor of the first.   Divide out common factors.   Add the two fractions. Simplify.

Lesson Quiz: Part I Add or subtract. Simplify your answer. 1. 2. 3. 4. 5.

Lesson Quiz: Part II 6. Vong drove 98 miles on interstate highways and 80 miles on state roads. He drove 25% faster on the interstate highways than on the state roads. Let r represent his rate on the state roads in miles per hour. a. Write and simplify an expression that represents the number of hours Vong drove in terms of r. b. Find Vong’s driving time if he averaged 55 miles per hour on the state roads. about 2 h 53 min