§ 6.2 Adding and Subtracting Rational Expressions

Adding Rational Expressions In this section, you will practice adding and subtracting rational expressions. Remember that when adding or subtracting fractions, it is necessary to rewrite the fractions as fractions having the same denominator, which is called the common denominator for the fractions being combined. Blitzer, Intermediate Algebra, 5e – Slide #2 Section 6.2

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, simplify the result. Blitzer, Intermediate Algebra, 5e – Slide #3 Section 6.2

Adding Rational Expressions EXAMPLE Add: SOLUTION This is the original expression. Add numerators. Place this sum over the common denominator. Combine like terms. Factor. Blitzer, Intermediate Algebra, 5e – Slide #4 Section 6.2

Adding Rational Expressions CONTINUED Factor and simplify by dividing out the common factor, x. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 6.2

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, simplify the result. Blitzer, Intermediate Algebra, 5e – Slide #6 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #7 Section 6.2

Subtracting Rational Expressions CONTINUED Combine like terms. Factor. Factor and simplify by dividing out the common factor, x + 3. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 6.2

Least Common Denominators EXAMPLE Find the LCD of: SOLUTION 1) Factor each denominator completely. 2) List the factors of the first denominator. Blitzer, Intermediate Algebra, 5e – Slide #10 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #12 Section 6.2

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, Blitzer, Intermediate Algebra, 5e – Slide #13 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #15 Section 6.2

Adding Fractions CONTINUED Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is, Blitzer, Intermediate Algebra, 5e – Slide #16 Section 6.2

Subtracting Fractions EXAMPLE Subtract: SOLUTION 1) Find the least common denominator. Begin by factoring the denominators. The factors of the first denominator are x - 4 and x – 1. The only factor from the second denominator that is unlisted is x + 1. Thus, the least common denominator is, Blitzer, Intermediate Algebra, 5e – Slide #17 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #18 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #19 Section 6.2

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, Blitzer, Intermediate Algebra, 5e – Slide #20 Section 6.2

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, Blitzer, Intermediate Algebra, 5e – Slide #21 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #22 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #23 Section 6.2

Addition of Fractions CONTINUED Since the numerator does not factor, there are clearly no common factors betwixt the numerator and the denominator. Therefore, the final solution is, Blitzer, Intermediate Algebra, 5e – Slide #24 Section 6.2

Addition of Fractions Add: This is the original expression. EXAMPLE Add: SOLUTION This is the original expression. Factor the first denominator. Multiply the numerator and the denominator of the second rational expression by -1. Multiply by -1. Blitzer, Intermediate Algebra, 5e – Slide #25 Section 6.2

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. Blitzer, Intermediate Algebra, 5e – Slide #26 Section 6.2

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, Blitzer, Intermediate Algebra, 5e – Slide #27 Section 6.2

Addition of Fractions Important to Remember Adding or Subtracting Rational Expressions If the denominators are the same, add or subtract the numerators and place the result over the common denominator. If the denominators are different, write all rational expressions with the least common denominator (LCD). Once all rational expressions are written in terms of the LCD, then add or subtract as described above. In either case, simplify the result, if possible. Even when you have used the LCD, it may be true that the sum of the fractions can be reduced. Blitzer, Intermediate Algebra, 5e – Slide #28 Section 6.2

Addition of Fractions Important to Remember Finding the Least Common Denominator (LCD) The LCD is a polynomial consisting of the product of all prime factors in the denominators, with each factor raised to the greatest power of its occurrence in any denominator. That is - After factoring the denominators completely, the LCD can be determined by taking each factor to the highest power it appears in any factorization. The Mathematics Teacher magazine accused the LCD of keeping up with the Joneses. The LCD wants everything (all of the factors) the other denominators have. Blitzer, Intermediate Algebra, 5e – Slide #29 Section 6.2