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Warm Up Add. Simplify your answer. 1. 2. 3.4. Subtract. Simplify your answer. 5. 6. 7. 8. Algebra 1B Chapter 11. Lesson Adding and Subtracting Rational Expressions. - PowerPoint PPT Presentation
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Warm UpAdd. Simplify your answer.
1. 2.
3. 4.
Subtract. Simplify your answer.
5.
7.
6.
8.
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.