Copyright © 2013, 2009, 2006 Pearson Education, Inc. Section 6.2 Adding and Subtracting Rational Expressions Copyright © 2013, 2009, 2006 Pearson Education, Inc. 1
Adding Rational Expressions With Common Denominators If are rational expressions, then To add rational expressions with the same denominator, add numerators and place the sum over the common denominator. If possible, factor and simplify the result.
Adding Rational Expressions EXAMPLE Add: SOLUTION This is the original expression. Add numerators. Place this sum over the common denominator. Combine like terms. Factor.
Adding Rational Expressions CONTINUED Factor and simplify by dividing out the common factor, x. Simplify.
Objective #1: Example
Objective #1: Example
Subtracting Rational Expressions With Common Denominators If are rational expressions, then To subtract rational expressions with the same denominator, subtract numerators and place the difference over the common denominator. If possible, factor and simplify the result.
Subtracting Rational Expressions EXAMPLE Subtract: SOLUTION This is the original expression. Subtract numerators. Place this difference over the common denominator. Remove the parentheses and distribute.
Subtracting Rational Expressions CONTINUED Combine like terms. Factor. Factor and simplify by dividing out the common factor, x + 3. Simplify.
Objective #2: Example
Objective #2: Example
Finding the Least Common Denominator (LCD) Least Common Denominators Finding the Least Common Denominator (LCD) 1) Factor each denominator completely. 2) List the factors of the first denominator. 3) Add to the list in step 2 any factors of the second denominator that do not appear in the list. 4) Form the product of each different factor from the list in step 3. This product is the least common denominator.
Least Common Denominators EXAMPLE Find the LCD of: SOLUTION 1) Factor each denominator completely. 2) List the factors of the first denominator.
Least Common Denominators CONTINUED 3) Add any unlisted factors from the second denominator. The second denominator is (2y – 1)(y + 4). One factor of y + 4 is already in our list, but the factor 2y – 1 is not. We add the factor 2y – 1 to our list. 4) The least common denominator is the product of all factors in the final list. Thus, is the least common denominator.
Objective #3: Example
Objective #3: Example
Objective #3: Example CONTINUED
Objective #3: Example
Objective #3: Example
Objective #3: Example CONTINUED
Add & Subtract Fractions Adding and Subtracting Rational Expressions That Have Different Denominators 1) Find the LCD of the rational expressions. 2) Rewrite each rational expression as an equivalent expression whose denominator is the LCD. To do so, multiply the numerator and denominator of each rational expression by any factor(s) needed to convert the denominator into the LCD. 3) Add or subtract numerators, placing the resulting expression over the LCD. 4) If possible, simplify the resulting rational expressions.
Adding Fractions EXAMPLE Add: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators. The factors of the first denominator are x + 4 and x – 2. The only factor from the second denominator that is unlisted is x – 1. Thus, the least common denominator is,
Adding Fractions CONTINUED 2) Write equivalent expressions with the LCD as denominators. This is the original expression. Factored denominators. Multiply each numerator and denominator by the extra factor required to form the LCD.
Adding Fractions CONTINUED 3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible. Add numerators. Perform the multiplications using the distributive property. Combine like terms.
Adding Fractions CONTINUED Since the numerator does not factor, there are clearly no common factors in the numerator and the denominator. Therefore, the final solution is,
Subtracting Fractions EXAMPLE Subtract: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators.
Subtracting Fractions CONTINUED 2) Write equivalent expressions with the LCD as denominators. This is the original expression. Factored denominators. Multiply each numerator and denominator by the extra factor required to form the LCD.
Subtracting Fractions CONTINUED 3) & 4) Add numerators, putting this sum over the LCD. Simplify, if possible. Subtract numerators. Perform the multiplications using the distributive property and FOIL. Remove parentheses.
Subtracting Fractions CONTINUED Combine like terms in the numerator. Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,
Addition of Fractions Perform the indicated operations: EXAMPLE Perform the indicated operations: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators. The factors of the first denominator are 1 and x – 3. The only factor from the second denominator that is unlisted is x + 1. We have already listed all factors from the third denominator. Thus, the least common denominator is,
Addition of Fractions CONTINUED 2) Write equivalent expressions with the LCD as denominator. This is the original expression. Factor the second denominator. Multiply each numerator and denominator by the extra factor required to form the LCD.
Addition of Fractions CONTINUED 3) & 4) Add and subtract numerators, putting this result over the LCD. Simplify if possible. Add and subtract numerators. Perform the multiplications using the distributive property. Combine like terms in the numerator.
Addition of Fractions CONTINUED Since the numerator does not factor, there are clearly no common factors in the numerator and the denominator. Therefore, the final solution is,
Objective #4: Example
Objective #4: Example
Objective #4: Example
Objective #4: Example
Objective #4: Example
Objective #4: Example
Objective #4: Example CONTINUED
Objective #4: Example CONTINUED
Objective #4: Example
Objective #4: Example
Objective #4: Example CONTINUED
Objective #4: Example CONTINUED
Addition of Fractions Add: This is the original expression. EXAMPLE Add: SOLUTION This is the original expression. Factor the first denominator.
Addition of Fractions Rewrite –y + x as x – y. CONTINUED Rewrite –y + x as x – y. Notice the LCD is (x + y)(x – y). Multiply the second numerator and denominator by the extra factor required to form the LCD. Perform the multiplications using the distributive property.
Addition of Fractions Add and subtract numerators. CONTINUED Add and subtract numerators. Remove parentheses and distribute. Combine like terms in the numerator. Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is,
Objective #5: Example
Objective #5: Example