RATIONAL

EXPRESSIONS

12 124

0 3 3

EVALUATING RATIONAL EXPRESSIONS

Evaluate the rational expression (if possible) for the given values of x: 12

3x

X = 0 X = 1 X = -3 X = 3

12 122

3 3 6

12 126

1 3 2

12 12

3 3 0undefined

DOMAIN OF A FUNCTIONThe domain of a function is the set of all real numbers that when substituted into the function produces a real

number

Steps to find the Domain of a Rational Expression

1. Set the denominator equal to zero and solve the resulting equation.

2. The domain is the set of real numbers excluding the values found in Step 1.

2y = -7

Find the Domain of Rational Expressions

3

2 7

y

y

2y + 7 = 0 Set the denominator equal to zero.

2y + 7 - = 0 - 7 Solve the equation

2 7

2 2

y 7

2y

Find the Domain of these Expressions

5

x

2

10

25

a

a

3

2

2 5

9

x

x

Simplifying a Rational Expression to Lowest Terms

2

2 14

49

p

p

1. Factor the numerator and denominator2. Determine the domain of the expression3. Simplify the expression to lowest terms

2

2 14 2( 7)1.

49 ( 7)( 7)

p p

p p p

2. (p + 7)(p – 7) = 0

p = -7 p = 7Domain: {p | p is a real number and p ≠ -7 and p ≠ 7

2

2 14 2( 7)1.

49 ( 7)( 7)

p p

p p p

Factor out the GCF in the numerator.

Factor the denominator as a difference of squares.

SOLUTION

2. (p + 7)(p – 7) = 0p + 7 = 0

p = -7 p = 7Domain: {p | p is a real number and p ≠ -7

and p ≠ 7

To find the domain restrictions, set the denominator equal to zero. The equation is

quadratic.

Set each factor equal to 0.

2( 7)3.( 7)( 7)

p

p p

Reduce common factors whose ratio is 1.

2

7p (provided p ≠ 7 or p ≠ -7)

Simplifying Rational Expressions to Lowest Terms

2

2 8

10 80 160

c

c c

SOLVE

3

3 2

8

2 4

x

x x

To reduce this rational expression, first factor the numerator and the denominator.

CAUTION:Remember when you

have more than one term, you cannot cancel with

one term. You can cancel factors only.

22 2 4x x x

22 2x x

2

2

2 4

2

x x

x

REDUCING RATIONAL EXPRESSIONS

There is a common factor so we can reduce.

Multiplication and Division of Rational Expressions

25 6

2 10

a b a

b

5 2 3

2 2 5

aab a

b

Factor into prime factors.

5 2 3

2 2 5

aab a

b

Simplify

33

2

a

2 3 2

2 2

6 8 3 2

4 1

x x x x x

x x x

To multiply rational expressions we multiply the numerators and then the denominators. However, if we can reduce, we’ll want to do that before combining so we’ll again factor first.

4 2x x

4x x

2 2

1

x x

x

MULTIPLYING RATIONAL EXPRESSIONS

2 1x x x

1 1x x

Now cancel any like factors on top with any like factors on bottom.

Now multiply numerator and multiply denominator

2 3 2

2 2

6 8 3 2

4 1

x x x x x

x x x

To divide rational expressions remember that we multiply by the reciprocal of the divisor (invert and multiply). Then the problem becomes a multiplying rational expressions problem.

4 2x x

4x x

2 2

1

x x

x

DIVIDING RATIONAL EXPRESSIONS

2 1x x x

1 1x x

2

2

2

3 2

6 841

3 2

x xx xx

x x x

Multiply by reciprocal of bottom fraction.

To add rational expressions, you must have a common denominator. Factor any denominators to help in determining the lowest common denominator.

ADDING RATIONAL EXPRESSIONS

2

2 2

3

4 12 4

x x

x x x

6 2x x 2 2x x

So the common denominator needs each of these factors.

6 2 2x x x

This fraction needs (x + 2)

This fraction needs (x + 6)

2

6 2

6

2

2 3x x

x

x

x

x

x

3 2 22 9 18

6 2 2

x x x x

x x x

3 23 9 18

6 2 2

x x x

x x x

FOILdistribute

Subtracting rational expressions is much like adding, you must have a common denominator. The important thing to remember is that you must subtract each term of the second rational function.

SUBTRACTING RATIONAL EXPRESSIONS

2

2 2

3

4 12 4

x x

x x x

6 2x x 2 2x x

So a common denominator needs each of these factors.

6 2 2x x x

This fraction needs (x + 2)

This fraction needs (x + 6)

2

6 2

6

2

2 3x x

x

x

x

x

x

-

3 2 22 9 18

6 2 2

x x x x

x x x

3 2 9 18

6 2 2

x x x

x x x

Distribute the negative to each term.

FOIL

2 9 18x x

For complex rational expressions you can combine the numerator with a common denominator and then do the same with the denominator. After doing this you multiply by the reciprocal of the denominator.

For complex rational expressions you can combine the numerator with a common denominator and then do the same with the denominator. After doing this you multiply by the reciprocal of the denominator.

COMPLEX RATIONAL EXPRESSIONS

3 12

1 32

xx

x

3 1 2

1 32

x xx

x

1

62

xxxx

For complex rational expressions you can combine the numerator with a common denominator and then do the same with the denominator. After doing this you multiply by the reciprocal of the denominator.

6

1 2x

x

x

x

For complex rational expressions you can combine the numerator with a common denominator and then do the same with the denominator. After doing this you multiply by the reciprocal of the denominator.

2 2

6

x

x