L05-Wed-14-Sep-2016-Sec-A-5-Rational-Expressions-HW04...

L05-Wed-14-Sep-2016-Sec-A-5-Rational-Expressions-HW04-Moodle-Q04, page 32

L05-Wed-14-Sep-2016-Sec-A-5-Rational-Expressions-HW04-Moodle-Q04

Rational Expressions are fractions where the numerator and denominator are polynomials.


We need to do 5 basic operations on rational expressions. We do these in the same way as we do for arithmetic fractions. This involves a lot of factoring.


A fraction or rational expression is in reduced form if the numerator and denominator have no common factors other than 1. In the following it is easy to see the common factor of 6:

18 3 3 3 324 2 2 2 2 2 2

6 6 16 46

In the following it is easy to see the common factor of 2:

204 102238 119

Are we done or are there more common factors?

Let's prime factor 102 and 119:

173 51

2 102 7 5 3 217

119

Now, we can see the additional common factor of 17.

2 2 12 2

204 • 2 • 3 • 217 17 117 1

3 2 3 6238 • 7 • 7 7 77

9 9 3

3 18 63 6x x x

x xx

Domain is | 6x x

What does it mean to reduce? The reduced version will give the same answer for all values in the domain of the original expression as would the original. The simplified version is easier to work with.

23 2

2

2 3 10 3 56 2 204 8 4 2 2

We can reduce by cancelling factors of , , and3 5to get . This is not identical to the original

2because the domains of the two rati

22

onal expressions

are not the sam

2

e. We

2

ca u

2

se

2

n

xx

x x x xx x xx x x x

xx

xx

x

3 2

2

3 5 in place of2

6 2 20 as long as we state the original domain4 8

| 0 , 2

x

x x xx x

x x x


Domain is | 0 , 4 , 4x x x x

Solution: 2

2

22 8 36 16x x

x x

4x

2 3 x3

x

4x

x reduces to

44x

xx

When we divide we want to change the denominator into a 1. For example:

Simplify:

2 2 6 2 62 63 3 5 3 5

5 5 6 1 3 56 6 5

A way to remember this is to recall "The Charge of the Light Brigade" an 1854 poem by Alfred, Lord Tennyson about the Battle of Balaclava during the Crimean War. In math we say Ours is not to reason why, just invert and multiply.

Simplify: 2

2

1 31 2

31

x xx x

x xx

2

21 3 1 11 2 3 1

x x x xx x xx x

3x

1 1

2

xxx

3x x

1 1

reduces to2

x xx x

Find the domain by noting where we get division by 0:

2

2

1 0 when 12 0 when 2

3 0 when 3 0 0 , 3

1 0 when 1 1 0 1 , 1

x xx x

x x x x x x

x x x x x

So, the domain is | 2, 1, 0, 1, 3x x


10 2 5 18 2 3 3 20 2 2 5 2 2 3 3 5 180LCD

1 5 3 1 18 5 10 3 9 18 50 27 4110 18 20 10 18 18 10 20 9 180 180

Note domain is | 3, 2,3x x

2 2

2

2 2 2

2 1 2 16 9 3 2 3 3

2 3 1 23 23 2 3 3

2 6 23 2 3 3 2 3

2 6 2 2 6 2 83 2 3 3 2 3 3 2 3

x xx x x x x x x

x x xx xx x x x

x x xx x x x x xx x x x x x x x

x x x x x x x x x


Simplify:

23

121

x

x

Note domain is 3| 0 , 1 ,2

x x

Solution:

2 3 2 3 23 3 2 13 2 111 2 1 1 2 2 1 2 3 2 32

1 1 1 1 1

x xx xx xx x x x

x x x x x xx x x x

We cannot reduce this. It is preferable to leave the fraction in factored form.

Let's do an example:

First, combine the fractions on the right into a single fraction. The LCD is 1 2R R

2 1 2 1

1 2 2 1 1 2

1 1R R R RR R R R R R

Now, the equation is

2 1

1 2

1 R RR R R

We can use a shortcut to get R by itself. Recall that if we have two fractions equal to each other (we call this a proportion) if we invert each fraction we end up with an equivalent equation. If we do that to this equation we have:

1 2

2 1

R RRR R

Below are the details as to why this works for a proportion:


Start with:

Multiply by LCD, which is

b

a cdb

bda

b ddc

db

We call this "cross multiplying"Divide by

d ca

a

bac

da

cc

b

a c

Thus, and are equivalent.

d

dc

ca

aa

b

bb d c

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