P.6 Rational Expressions EX: Find all the numbers that must be excluded from the domain of each rational

expression.

a. y b. x + 1 x-2 x2 - 1

SolutionTo determine the numbers that must be excluded from each domain, examine the denominators. You must _EXCLUDE_from the

DOMAIN any value for the variable(s) that would make the

_DENOMINATOR_ (before reducing) = ___ZERO______________.

a. b.2 or (- ,2) (2, )x 1 or (- ,-1) (-1,1) (1, )x

Simplifying Rational Expressions

1. Factor the numerator and denominator completely.

2. Reduce (Divide both the numerator and denominator by the common factors. Remember a factor is something that is being MULTIPLIED!)

84

42

x

x• Ex: Simplify:

Solution (note the form on simplifying the expression):

2 4 ( 2)( 2)

4 8 4( 2)

x+2 =

4

x x x

x x

Q: What is the domain? Ans: 2 or (- ,2) (2, )x

You CANNOT reduce by dividing out 2 as 2 is not a FACTOR (multiplied) in both.

Multiplying Rational Expressions

1. FACTOR all numerators and denominators completely.

2. REDUCE (Divide both the numerator and denominator by common factors.)

3. MULTIPLY the remaining factors in the numerator and multiply the remaining factors in the denominator.

2 2

2 2

2 3 1 (2 3) ( 1)( 1)

2 3 2 (2 3)( 1) ( 2)

1

2

: 2, 1,0,3/ 2

x x x x x x x

x x x x x x x x

x

x

Domain x

• EX: Multiply and simplify:

When we divide rational expressions, we “keep, change, flip” so that our result is in the form of a multiplication problem, so we can apply the previous rules. Example:

2

3

246

63 2

2

2

x

xx

x

xx• EX: Divide and simplify:2 2

2

2

2 2

3 6 3

6 24 2

3 6 2

6 24 3

x x x x

x x

x x x

x x x

66

1

1

1

6

1

)1(3

2

)2)(2(6

)2(3

xx

xx

x

xx

xx

To ADD ( or subtract) rational expressions, we do the same thing we would for a fraction: Add the numerators, and keep the denominators exactly the same. (Then reduce if possible).Example:

13

3

13

2

xx

x• Add:

Solution:

13

32

13

3

13

2

x

x

xx

x

Finding the Least Common Denominator

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.

(If needed will do a brief example.)

Adding and Subtracting Rational Expressions That Have Different

Denominators with Shared Factors1. LCD: Find the least common denominator.2. EQUIVALENT EXPRESSIONS: Write all rational

expressions in terms of the LCD. To do so, multiply both the numerator and the denominator of each rational expression by any factor(s) needed to convert the denominator into the least common denominator.

3. ADD or subtract the numerators, placing the resulting expression over the least common denominator.

4. REDUCE: If necessary, simplify the resulting rational expression.

55

2

55

42

aaa

• EX: Subtract:

Solution:

)1(5

2

)1(5

455

2

55

42

aaa

aaa

)1(5

24

)1(5

2

)1(5

4

)1(5

2

)1(5

4

aa

a

aa

a

aa

a

a

aaa

LCD (factor and find)

Equivalent expressions

Add numerator/(keep denominator same)

Reduce if possible

Simplifying “COMPLEX” rational expressions

• We will take out the fraction within a fraction by multiplying both the numerator AND denominator by the LCD of all terms in the numerator and denominator. (See ex 9 p 67 for alternative method.)

• Ex: Simplify1

42

xxx

LCD?

14

42 4

xx

xx x

2 4

4 (2 )

( 2)( 2)

4 (2 )

2

4

x

x x

x x

x x

x

x

Don’t hesitate to ask about this type of problem!