Warm UpAdd. Simplify your answer.

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Subtract. Simplify your answer.

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Algebra 1BAlgebra 1BChapter 11Chapter 11

Lesson Adding and Subtracting Rational Expressions

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.

Example 1C: Adding Rational Expressions with Like Denominators

Add. Simplify your answer.

Combine like terms in the numerator.

Factor. Divide out common factors.

Simplify.

In Your Notes! Example 1a

Add. Simplify your answer.

= 2

Combine like terms in the numerator. Divide out common factors.

Simplify.

In Your Notes! Example 1b

Add. Simplify your answer.

Combine like terms in the numerator.

Factor. Divide out common factors.

Simplify.

Example 2: Subtracting Rational Expressions with Like Denominators

Subtract. Simplify your answer.

Subtract numerators.

Combine like terms.

Factor. Divide out common factors.

Simplify.

In Your Notes! Example 2a

Subtract. Simplify your answer.

Subtract numerators.

Combine like terms.

Factor. Divide out common factors.

Simplify.

In Your Notes! 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 any common multiple of the denominators to find one. You can also use the least common multiple (LCM) of the denominators.

To find the LCM of two expressions, write the prime factorization of both expressions. Line up the factors as shown. To find the LCM, multiply one number from each column.

Example 3A: Identifying the Least Common Multiple

Find the LCM of the given expressions.

12x2y, 9xy3

12x2y = 2 2 3 x x y

9xy3 = 3 3 x y y y

LCM = 2 2 3 3 x x y y y

Write the prime factorization of each expression. Align common factors. = 36x2y3

Example 3B: Identifying the Least Common Multiple

Find the LCM of the given expressions.

c2 + 8c + 15, 3c2 + 18c + 27

c2 + 8c + 15 = (c + 3) (c + 5)

3c2 + 18c + 27 = 3(c2 + 6c +9)

= 3(c + 3)(c + 3)

LCM = 3(c + 3)2(c + 5)

Factor each expression.

Align common factors.

In Your Notes! Example 3a

Find the LCM of the given expressions.

5f2h, 15fh2

5f2h = 5 f f h

15fh2 = 3 5 f h h

LCM = 3 5 f f h h

= 15f2h2

Write the prime factorization of each expression. Align common factors.

In Your Notes! Example 3b

Find the LCM of the given expressions.

x2 – 4x – 12, (x – 6)(x + 5)

x2 – 4x – 12 = (x – 6) (x + 2)

(x – 6)(x + 5) = (x – 6)(x + 5)

LCM = (x – 6)(x + 5)(x + 2)

Factor each expression.

Align common factors.

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

Adding or Subtracting Rational Expressions

Step 1 Identify a common denominator.

Step 3 Write each expression using the common denominator.

Step 2 Multiply each expression by an appropriate form of 1 so that each term has the common denominator as its denominator.

Step 4 Add or subtract the numerators, combining like terms as needed.

Step 5 Factor as needed.

Step 6 Simplify as needed.

Example 4A: Adding and Subtracting with Unlike Denominators

Add or subtract. Simplify your answer.

Step 15n3 = 5 n n n2n2 = 2 n nLCD = 2 5 n n n = 10n3

Identify the LCD.

Step 2Multiply each expression

by an appropriate form of 1.

Write each expression using the LCD.

Step 3

Example 4A Continued

Add or subtract. Simplify your answer.

Add the numerators.

Factor and divide out common factors.

Step 6 Simplify.

Step 4

Step 5

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

Step 3

Multiply the first expression

by to get an LCD of

w – 5. Write each expression

using the LCD.

Example 4B Continued

Add or Subtract. Simplify your answer.

Step 4

Step 5, 6

Subtract the numerators.

No factoring needed, so just simplify.

Add or subtract. Simplify your answer.

Identify the LCD.3d 3 d 2d3 = 2 d d d

LCD = 2 3 d d d = 6d3 Step 1

Multiply each expression by an appropriate form of 1.

Write each expression using the LCD.

In Your Notes! Example 4a

Step 2

Step 3

Add or subtract. Simplify your answer.

In Your Notes! Example 4a Continued

Subtract the numerators.

Factor and divide out common factors.

Step 4

Simplify.

Step 5

Step 6

Add or subtract. Simplify your answer.In Your Notes! Example 4b

Factor the first term. The denominator of second term is a factor of the first.

Add the two fractions.

Divide out common factors.

Step 1

Step 4 Simplify.

Step 2

Step 3

Lesson Quiz 11.5

Add or subtract. Simplify your answer.

1. 2.

5.

3. 4.