RATIONAL

EXPRESSIONS

Definition of a Rational Expression

A rational number is defined as the ratio of two integers, where q ≠ 0

Examples of rational numbers:

p

q

2 1, ,93 5

A rational expression is defined as the ratio of two polynomials, where q ≠ 0.

Examples of rational expressions:

p

q

2

2

3 6 3 6 2, ,4 4 7

x r r

x

Domain of a Rational Expression

Domain of a Rational ExpressionThe domain of a Rational Expression is the set of all real numbers that when substituted into the Expression produces a real number.

3

2

x

x

If you choose x = 2, the denominator will be 2 – 2 = 0 which is illegal because you can't divide by zero. The answer then is:

{x | x 2}.

illegal if this is zero

Note: There is nothing wrong with the top = 0 just

means the fraction = 0

Finding the Domain of Rational Expression

2

10

25

a

a

Set the denominator equal to zero. The equation is quadratic.

2 25 0a Factor the equation

( 5)( 5) 0a a Set each factor equal to zero.Solve

5 0a 5 0a The domain is the set of real numbers except 5 and -5

Domain: {a | a is a real number and a ≠ 5, a ≠ -5}

5a 5a

2

2 14

49

p

p

2( 7)p

( 7)( 7)p p

REDUCING RATIONAL EXPRESSIONS

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

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

( 7)( 7) 0p p Set each factor equal to 0.

7 0p 7 0p

p = -7 or p = 7

The domain is all real numbers except -7 and 7.

2

2 14

49

p

p

CAUTION:Remember when you

have more than one term, you cannot cancel with

one term. You can cancel factors only.

2( 7)p

( 7)( 7)p p

2

7p

REDUCING RATIONAL EXPRESSIONS

There is a common factor so we can reduce.

Provided p ≠ 7 and p ≠ -7

2

2 8

10 80 160

c

c c

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.

2( 4)c

2 5( 4)( 4)c c

1

5( 4)c

REDUCING RATIONAL EXPRESSIONS

There is a common factor so we can reduce.

1

Simplifying a Ratio of -1

51

5

The ratio of a number and its opposite is -1

21

2

x

x

2 1( 2 ) 1( 2) 1

12 ( 2) ( 2) 1

x x x

x x x

factor out a -1

3 3c d

d c

Simplifying Rational Expressions to Lowest Terms.

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

Reduce common factors to lowest terms.

33

1

3( )c d

Notice that (c – d) and (d – c) are opposites and form a ratio of -1

1( )c d

Solution

2

5

25

y

y

Simplifying Rational Expressions to Lowest Terms.

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

Reduce common factors to lowest terms.

1

5y

1( 5)y

Notice that (y - 5) and (5 – y) are opposites and form a ratio of -1

( 5)( 5)y y

Solution

Multiplication of Rational Expressions

Multiplication of Rational ExpressionsLet p, q, r, and x represent polynomials, such that q ≠ 0 s ≠ 0. Then,

p r pr

q s qs

2 2

3 3 2

6

c d

c c d

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.

3( )c d

2 3 c

1

( )c c d

MULTIPLYING RATIONAL EXPRESSIONS

c d c d

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

Simplify.

1

1

Division of Rational Expressions

Division of Rational ExpressionsLet p, q, r, and x represent polynomials, such that q ≠ 0 s ≠ 0. Then,

p r p s ps

q s q r qr

2

5 5 10

2 9

t

t

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.

5( 3)t

25

3t

DIVIDING RATIONAL EXPRESSIONS

2 5

( 3)( 3)t t

2

5 1529

10

t

t

Multiply by reciprocal of bottom fraction.

1

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.

DIVIDING RATIONAL EXPRESSIONS

2 2

2

11 30 30 5

10 250 2 4

p p p p

p p

Multiply by reciprocal of bottom fraction.

2

2 2

11 30 2 4

10 250 30 5

p p p

p p p

Factor

( 5)( 6)p p 2( 2)p

2 5( 5)( 5)p p 5 (6 )p p

Notice that (p - 6) and (6 – p) are opposites and form a ratio of -1

5 ( 6)p p Reduce common factors

( 2)

25 ( 5)

p

p p

Solution

Addition and Subtraction of Rational Expressions

Addition and Subtraction of Rational Expressions

Let p, q, and r represent polynomials where q ≠ 0. Then,

1.

2.

Addition and Subtraction of Rational Expressions

Let p, q, and r represent polynomials where q ≠ 0. Then,

1.

2.

p r p r

q q q

p r p r

q q q

Adding and Subtracting Rational Expressions with a Common

Denominator

1 7

12 12 The fractions have the same denominator.

Add term in the numerators, and write the result over the common denominator.

1 7

12

8

12

Simplify to lowest terms.

2

3 Solution

Adding and Subtracting Rational Expressions with a Common

Denominator2 5 24

3 3

x x

x x

The fractions have the same denominator.

Subtract the terms in the numerators, and write the result over the common denominator.

2 5 24

3

x x

x

Simplify the numerator.

Factor the numerator and denominator to determine if the rational expression can be simplified.

2 5 24

3

x x

x

( 8)( 3)

3

x x

x

Simplify to lowest terms.8x

Steps to Add or Subtract Rational Expressions

• Factor the denominators of each rational expression.

• Identify the LCD• Rewrite each rational expression as an

equivalent expression with the LCD as its denominator.

• Add or subtract the numerators, and write the result over the common denominator.

• Simplify to lowest terms.

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

1 10

5 25x x

5 5x x ( 5)x

So the common denominator needs each of these factors.

This fraction needs (x + 5)

This fraction needs nothing

101

5

5

5x x

x

5

5 5

x

x x

simplifydistribute

Reduce common factors1

1

( 5)x Solution

5 10

5 5

x

x x

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

ADDING RATIONAL EXPRESSIONS

1 5

7 7d d

The expressions d - 7 and 7- d are opposites and differ by a factor of -1 Therefore, multiply the numerator and denominator of either expression by -1 to obtain a common denominator.

1 ( 5)

7d

Simplify

Solution

1 5( 1)

( 1)7 (7 )d d

4

7d

Add the terms in the numerators, and write the result over the common denominator.

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

SUBTRACTING RATIONAL EXPRESSIONS

2 4 1

3 2

q q

23

So the common denominator needs each of these factors.The LCD is 6.

This fraction needs (2)

This fraction needs (3)

2 3(2 14 )

6

qq

11

6

q

simplifydistribute

Reduce common factors

Solution

4 8 3 3

6

q q

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