Warm Up Add Simplify your answer 1 2
- Slides: 27
Warm Up Add. Simplify your answer. 1. 2. 3. 4. Subtract. Simplify your answer. 5. 6. 7. 8.
Algebra 1 B Chapter 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 1 A: Adding Rational Expressions with Like Denominators Add. Simplify your answer. Combine like terms in the numerator. Divide out common factors. Simplify.
Example 1 B: Adding Rational Expressions with Like Denominators Add. Simplify your answer. Combine like terms in the numerator. Factor. Divide out common factors. Simplify.
Example 1 C: 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 1 a Add. Simplify your answer. Combine like terms in the numerator. Divide out common factors. =2 Simplify.
In Your Notes! Example 1 b 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 2 a Subtract. Simplify your answer. Subtract numerators. Combine like terms. Factor. Divide out common factors. Simplify.
In Your Notes! Example 2 b 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 3 A: Identifying the Least Common Multiple Find the LCM of the given expressions. 12 x 2 y, 9 xy 3 = 3 3 x y y y Write the prime factorization of 12 x 2 y = 2 2 3 x x y each expression. LCM = 2 2 3 3 x x y y y Align common factors. = 36 x 2 y 3
Example 3 B: Identifying the Least Common Multiple Find the LCM of the given expressions. c 2 + 8 c + 15, 3 c 2 + 18 c + 27 = 3(c 2 + 6 c +9) = 3(c + 3) Factor each expression. (c + 5) Align common factors. LCM = 3(c + 3)2(c + 5) c 2 + 8 c + 15 = (c + 3)
In Your Notes! Example 3 a Find the LCM of the given expressions. 5 f 2 h, 15 fh 2 5 f 2 h = 5 f f h 15 fh 2 = 3 5 f h h LCM = 3 5 f f h h = 15 f 2 h 2 Write the prime factorization of each expression. Align common factors.
In Your Notes! Example 3 b Find the LCM of the given expressions. x 2 – 4 x – 12, (x – 6)(x + 5) Factor each x 2 – 4 x – 12 = (x – 6) (x + 2) expression. Align common (x – 6)(x + 5) = (x – 6)(x + 5) factors. LCM = (x – 6)(x + 5)(x + 2)
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 2 Multiply each expression by an appropriate form of 1 so that each term has the common denominator as its denominator. Step 3 Write each expression using the common denominator. Step 4 Add or subtract the numerators, combining like terms as needed. Step 5 Factor as needed. Step 6 Simplify as needed.
Example 4 A: Adding and Subtracting with Unlike Denominators Add or subtract. Simplify your answer. Identify the LCD. 5 n 3 = 5 n n n Step 1 2 n 2 = 2 n n LCD = 2 5 n n n = 10 n 3 Step 2 Step 3 Multiply each expression by an appropriate form of 1. Write each expression using the LCD.
Example 4 A Continued Add or subtract. Simplify your answer. Step 4 Add the numerators. Step 5 Factor and divide out common factors. Step 6 Simplify.
Example 4 B: 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. Multiply the first expression Step 2 by to get an LCD of Step 3 w – 5. Write each expression using the LCD.
Example 4 B Continued Add or Subtract. Simplify your answer. Step 4 Subtract the numerators. Step 5, 6 No factoring needed, so just simplify.
In Your Notes! Example 4 a Add or subtract. Simplify your answer. 3 d 3 d Step 1 2 d 3 = 2 d d d Identify the LCD = 2 3 d d d = 6 d 3 Step 2 Multiply each expression by an appropriate form of 1. Step 3 Write each expression using the LCD.
In Your Notes! Example 4 a Continued Add or subtract. Simplify your answer. Step 4 Subtract the numerators. Step 5 Factor and divide out common factors. Step 6 Simplify.
In Your Notes! Example 4 b Add or subtract. Simplify your answer. Step 1 Factor the first term. The denominator of second term is a factor of the first. Step 2 Add the two fractions. Step 3 Divide out common factors. Step 4 Simplify.
Lesson Quiz 11. 5 Add or subtract. Simplify your answer. 1. 2. 3. 4. 5.
