RATIONAL

EXPRESSIONS

Rational Expressions and Functions: Multiplying and Dividing

• Objectives– Simplifying Rational Expressions and

Functions– Rational Functions– Multiplying– Dividing and Simplifying

Definition of a Rational Expression

An expression that consists of a polynomial divided by a nonzero polynomial is called a rational expression.

Examples of rational numbers: 2 1 9

, , ,3 5 a b

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

Rational Functions

Like polynomials, certain rational expressions are used to describe

functions. Such functions are called rational functions.

The function given by2 5

( )2 5

t tH t

t

Gives the time, in hours, for two machines, working together, to complete a job that the first machine could do alone in t hours and the second machine could do in t + 5 hours. How long will the two machines, working together, require for the job if the first machine alone would take (a) 1 hour? (b) 6 hours

Solution 2

2

5 1 5 6) ( )

2 5 2 5 7

5 36 30 66 15) ( ) 3

2

1 11

1

6 66

6 5 12 5 17 17

a H hr

b H or hr

Since division by 0 is undefined, the domain of a rational function must exclude any numbers for which the denominator is 0.

The domain of H is 5 5( , ) ( , )

2 2

Multiplying

Products of Rational ExpressionsTo multiply two rational expressions, multiply numerators and multiply denominators:

p r pr

q s qs

Recall from arithmetic that multiplication by 1 can be used to find equivalent expressions:

23

2

3

5 56

10

Multiplying by 2/2, which is 1

3/5 and 6/10 represent the same number

Similarly, multiplication by 1 can be used to find equivalent rational expressions:

3

3

5 5

2 2( 5)( 3)

( 2)( 3)

x

x

x xand

x xx x

x x

Multiplying by , which is 1,

Provided x ≠ -3

3

3

x

x

So long as x is replaced with a number other than -2 or -3, the expressions

represent the same number. If x = 4, then

and

45 3 ( 5)( 3)

2 3 ( 2)( 3

4

) 424

1

64

7x x

x x

5 5 3

2 2 3

x x xand

x x x

45 5

2

1

64 2

xand

x

A rational expression is said to be simplified when no factors equal to 1 can be removed.

Simplifying Rational Expressions and Functions

We “removed” the factor that equals 1 :

It is important that a rational function’s domain not be changed as a result of simplifying

There is a serious problem with stating that the functions

31

3

x

x

( 5)( 3) 5 5

( 2)(

3

3) 232

x x x xan

x

xd

x x x x

( 5)( 3) 5( ) ( )

( 2)( 3) 2

x x xF x and G x

x x x

( 5)( 3) 5

( 2)( 3) 2

x x x

x x x

represent the same function. The difficulty arises from the fact that the domain of each function is assumed to be all real numbers for which the denominator is nonzero. Thus, unless we specify otherwise,

There is a serious problem with stating that the functions

( 5)( 3) 5( ) 3

( 2)( 3) 2

x x xF x with x

x x x

represent the same function. The difficulty arises from the fact that the domain of each function is assumed to be all real numbers for which the denominator is nonzero. Thus, unless we specify otherwise,

Domain of F = { x ≠ -2, x ≠ -3}, andDomain of G = { x| x ≠ -2}

Thus, as presently written, the domain of G includes -3, but the domain of F does not. This problem is easily addressed by specifying

( 5)( 3) 5( ) ( )

( 2)( 3) 2

x x xF x and G x

x x x

Write the function given byin simplified form.

27 21( )

14

t tf t

t

Solution We first factor the numerator and the denominator, looking for the largest factor common to both. Once the greatest common factor is found, we use it to write 1 and simplify:

27 21( )

14( 3)

27 3

7 23,

7

7

02

3, 0

2

t tf t

tt

t t

tt

t

tThus simplified formis f t witht

t

t

Note that the domain of f = {t |t ≠ 0}

Factoring. The greatest common factor is 7t

Write the function given byin simplified form , and list all restrictions on the domain.

2

2

3 10g( )

2 3 2

x xt

x x

Solution

Note that the domain of g = {x |x ≠ -1/2 and 2}

Factoring the numerator and the denominator.

2

2

3 10g( )

2 3 2

( 2)( 5)

(2 1)( 2)

2 5

2 2 1

5 1, ,2

2 1 25 1

( ) , , 22 1 2

x xt

x x

x x

x x

x x

x x

xx

xx

thus g x xx

Rewriting as a product of two rational expressions

Remove a factor equal to 1:

21

2

x

x

Let’s try:

2 2

2 2

9 6 3)12 12

4)3 12

x xy ya

x y

tb

t

2 2

3 3 2

6

c d

c c d

“Canceling” is a shortcut often used for removing a factor equal to 1 when working with fractions. Canceling removes factors equal to 1 in products. It cannot be done in sums or when adding expressions together.

3( )c d

2 3 c

1

( )c c d

CANCELING

c d c d

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

Simplify.

1

1

1

1 1

Multiply. Then simplify by removing a factor equal to 1

2 2

2 2

2 4 ( 2)( 4)

3 2 ( 3)( 2)

( 2)( 2)

( 3) (x

( 2)

(x 2) 1)

(x 2)(x 2)(x 2)

( 3)( 2)( 1)

x x x x

x x x x x x

x x

x

x x x

x

Factoring the numerator and the denominator and finding common factors

Multiplying the numerators and also the denominators

Remove a factor equal to 1

( 2)( 2)

( 3)( 1)

x x

x x

Simplify

Let’s try:

3 5

2 2

1)

1

a aa

a a

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.

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