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Section 6.2 A

Adding and Subtracting Rational Expressions

First, let’s review adding and subtracting rational numbers.

Adding fractions and subtracting fractions both require finding a least common denominator (LCD), which is most easily done by factoring the denominator (bottom number) of each fraction into a product of prime numbers (a number that can be divided only by itself and 1.)

4

Example: Adding Fractions

Step 1: Factor the two denominators into prime factors, then write each fraction with its denominator in factored form:

10 = 2 5 ∙ and 35 = 5 7∙

so 3 + 2 = 3 + 2 10 35 2 5 ∙ 5 7∙

Step 2: Find the least common denominator (LCD): LCD = 2 5∙ 7∙

5

Step 3: Multiply the numerator (top)and denominator of each fraction by the factor(s) it’s missing from the LCD.

LCD = 2∙5 7∙ 3∙7 + 2 2 ∙ . 2 5∙ 7∙ 5 7∙ 2∙

Step 4: Multiply each numerator out, leaving the denominators in factored form, then add the two numerators and put them over the common denominator. 21 + 4 = 21 + 4 = 25 2 5 7 5 7 2 2 5 7 2 5 7∙ ∙ ∙ ∙ ∙ ∙ ∙ ∙

Step 5: Now factor the numerator, then cancel any common factors that appear in both numerator and denominator. Once you multiply out any remaining factors, the result is your simplified answer.

= 25 = 5 5 ∙ = 5 5 ∙ = 5 = 5 . 2 5 7 2 5 7 2 5 7 2 7 14∙ ∙ ∙ ∙ ∙ ∙ ∙

/ / 6

Rational expressions can be added or subtracted in the same way as fractions:

1. Find a least common denominator by factoring the polynomials in each denominator.

2. Multiply each rational expression top and bottom by the missing factor(s) in the LCD.

3. Multiply out the polynomials in each numerator, but leave the denominators as they are (in factored form).

4. Add or subtract the resulting numerator polynomials and put the result over the factored LCD polynomial.

5. Simplify if needed, by seeing if you can factor the numerator polynomial, then cancelling any common factors.

Add the following rational expressions.

72

83

72

34

p

p

p

p72

57

p

p

Example

72

8334

p

pp

Note: If the denominators are already the same, you don’t need to factor and find an LCD.

Subtract the following rational expressions.

2

16

2

8

yy

y

2

168

y

y

2

)2(8

y

y8

Example

Subtract the following rational expressions.

103

6

103

322 yyyy

y

103

632 yy

y

)2)(5(

)2(3

yy

y

5

3

y

Example

To add or subtract rational expressions with unlike denominators, you have to change them to equivalent forms that have the same denominator (a common denominator).

This involves finding the least common denominator of the two original rational expressions.

To find a Least Common Denominator

1) Factor the given denominators.

2) Take the product of all the unique factors.

Factors that appear in more than one denominator are not included more than once in the LCD unless they are repeated more than once in any ONE denominator.

Do not multiply out the LCD factors (other than any pure number factors) – leave the denominator in factored form. e.g. leave (x+2)(x+3) as is, not as x2 + 5x + 6.

Find the LCD of the following rational expressions.

124

3,

6

1

y

x

y

6 2 3y y 4 12 4( 3 2 2( 3))y y y

So the LCD is 2 ( 12 ( 3)2 3 3)y y y y

Example

Find the LCD of the following rational expressions.

2110

24,

34

422

xx

x

xx2 ( 3)( 14 3 )x xx x

2 ( 3)( 7)10 21 xx xx

(xSo (x 73the LC )(xD s )1) i

Example

Problem from today’s homework:

Find the LCD of the following rational expressions.

12

4,

55

32

2

2 xx

x

x

x

2 25 5 5( 1) 5( 1)( 1)xx xx 2 ( 1)( 12 1 )x xx x

25(x 1)(x - (x - 1)So the LCD is = 5(x+1)1) (x-1)

Example

Problem from today’s homework:

Find the LCD of the following rational expressions.

xx 3

2,

3

1

Neither denominator can be factored further.Since x – 3 = –(3 – x), you can use x – 3 as the LCD.

(Note: The product (x – 3)(3 – x) would work as a common denominator, but it would not be the smallest or least common denominator.)

Example

Rewrite the rational expression as an equivalent rational expression with the given denominator.

95 729

3

yy

59

3

y

4

4

5 8

8

9

3

y

y

y 9

4

72

24

y

y

Example

5 9Q: 9 2? =7y y

5 948A: 9 = 2y 7y y

Adding or Subtracting Rational Expressions with Unlike Denominators

1) Find the LCD of all the rational expressions.

2) Rewrite each rational expression as an equivalent one with the LCD as the denominator.

3) Multiply out each numerator, then add or subtract and write result over the LCD.

4) Simplify rational expression, if possible.

Problem from today’s homework:

Problem from today’s homework:

Problem from today’s homework:

Problem from today’s homework:

Reminder:This homework assignment

on section 6.2A is due at the start of

next class period.

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