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Rational Expressions and Common Denominators

In order to perform addition and subtraction operations with rational numbers, it is necessary to convert all
numbers so that they have a common denominator.

For example, the operation cannot be performed without converting to

To add or subtract rational numbers or expressions, you must have a common denominator. Once both terms
have the same denominator, they can be added or subtracted easily by adding the numerators.

A common denominator is generally chosen to be the least common multiple of the denominators. For rational
numbers and rational expressions, this can be thought of as the smallest number or expression that is divisible
by every term.

EX: Find the least common multiple of 16 and 40.

The standard way for finding the least common multiple of two integers involves computing the prime
factorization of each. Breaking down 16 and 40 results in the following:

16 = 2 * 2* 2 * 2 and 40 = 2 * 2* 2 * 5

It may be helpful to list the factors in a table form in order to carry out the process. We also include a
column for factors that we will add to the least common multiple (this column will be empty at first).

Factors of 16	Factors of 40	Least Common Multiple
2, 2, 2, 2	2, 2, 2, 5

The next step is to move all factors that are common to both 16 and 40 over to the least common
multiple column. In this case, there are three 2's that are common to both. Move these over, but
remember to only place them in the least common multiple column once.

Factors of 16	Factors of 40	Least Common Multiple
2	5	2, 2, 2

Now move over all remaining factors of the first number.

Factors of 16	Factors of 40	Least Common Multiple
	5	2, 2, 2

And then the remaining factors of the second number.

Factors of 16	Factors of 40	Least Common Multiple
		2, 2, 2,5

Finally, multiply out all of the least common multiple factors. The least common multiple is therefore
given as 2 * 2 * 2* 2 * 5 = 80 .

EX: Find the least common multiple of 30, 24, and 32.

Finding the least common multiple of three integers is slightly more complicated, but the process is very
similar.

Factors of 30	Factors of 24	Factors of 32	Least Common Multiple
2, 3, 5	3, 3, 2, 2	2, 2, 2, 2, 2

First move over all factors that are common to all three. In this case, a single 2 is the only one.

Factors of 30	Factors of 24	Factors of 32	Least Common Multiple
3, 5	3, 3, 2	2, 2, 2, 2	2

Then move over all factors that are common to the first and second numbers. Only a single 3 qualifies.

Factors of 30	Factors of 24	Factors of 32	Least Common Multiple
5	3, 2	2, 2, 2, 2	2,3

Move over all factors common to the first and third. There are none.

Next move over all factors common to the second and third. Only a single 2 qualifies.

Factors of 30	Factors of 24	Factors of 32	Least Common Multiple
5	3	2, 2, 2	2,3, 2

Go in order, moving over the remaining factors from each of the three numbers.

Factors of 30	Factors of 24	Factors of 32	Least Common Multiple
	3	2, 2, 2	2,3, 2, 5
Factors of 30	Factors of 24	Factors of 32	Least Common Multiple
		2, 2, 2	2, 3, 2, 5, 3
Factors of 30	Factors of 24	Factors of 32	Least Common Multiple
			2, 3, 2, 5, 3, 2, 2, 2

Finally, multiply the factors out to get the least common multiple:

2 * 3* 2* 5* 3* 2 * 2 * 2 = 2* 2 * 2 * 2 * 2* 3* 3* 5 = 1440

Thus the least common multiple is 1440.

Notice that the factorization of the least common multiple completely contains the factorization of each
of the original three numbers, but with some overlap.

EX: Find the least common multiple of

For expressions with variables and exponents, it is very simple to find the least common multiple. For
each variable, take the highest exponent that appears in either expression.

For x, the highest exponent is 5, in the second expression.
For y, the highest exponent is 3, in the first expression.
For z, the highest exponent is 4, in the second expression.

Therefore the least common multiple is

EX: Find the least common multiple of x^2 + 5x + 6 and x^2 - 9 .

Just like with integers, the first step towards finding a least common multiple of polynomials is to factor
each polynomial.

Factors of x^2 + 5x + 6	Factors of x^2 - 9	Least Common Multiple
(x + 2) , (x + 3)	(x + 3) , (x - 3)

Move the common factors over. In this case, only (x + 3) is common.

Factors of x^2 + 5x + 6	Factors of x^2 - 9	Least Common Multiple
(x + 2)	(x - 3)	(x + 3)

Now move over the remaining factors from each expression.

Factors of x^2 + 5x + 6	Factors of x^2 - 9	Least Common Multiple
	(x - 3)	(x + 3)(x + 2)

Factors of x^2 + 5x + 6	Factors of x^2 - 9	Least Common Multiple
		(x + 3)(x + 2)(x - 3)

Therefore the least common multiple of x^2 + 5x + 6 and x^2 - 9 is (x + 3)(x + 2)(x - 3)

Adding Rational Expressions

The following steps are necessary when adding rational expressions:

1) Find a common denominator. This is usually the least common multiple of the denominators.
2) Convert all addends so that they have the same denominator.
3) Add the numerators and place the result over the common denominator.
4) If necessary, simplify the numerator.
5) If necessary, factor the numerator and simplify the rational expression.

EX: Simplify

The first step is to find the least common multiple of (y^2)z and xy so that it can be used as a common
denominator. Simply take the highest exponent of each variable. The highest for x is 1 (second
expression); the highest for y is 2 (first expression); the highest for z is 1 (first expression).

Thus the common denominator will be x(y^2)z .

Now find out what factors are missing from each denominator that will make them equal to the common
denominator.

The first denominator is missing a single x, so multiply the first expression by x/x, which is equal to 1.

The second denominator is missing yz , so multiply the second expression by ,also equal to 1.

Now we get the following:

	[Now simplify the two multiplications.]
	[Add the numerators, placing the sum over the common denominator.]
	[The numerator cannot be simplified or factored.]

EX: Simplify

In this case, both addends have the same denominator. Therefore the numerators can immediately be
added together.

[This cannot be further simplified.]

EX: Simplify

Here we must first find the least common multiple of 6x and 8x.

Factors of 6x	Factors of 8x	Least Common Multiple
2, 3, x	2, 2, 2, x

The common factors are x and a single 2.

Factors of 6x	Factors of 8x	Least Common Multiple
3	2, 2	2, x

Move over the remaining factors for each.

Factors of 6x	Factors of 8x	Least Common Multiple
	2, 2	2, 3, x

Factors of 6x	Factors of 8x	Least Common Multiple
		2, 2, 2, 3, x

Therefore the common denominator will be (2 * 2 * 2 * 3)x = 24x .

Now find out what factors are missing from each denominator that will make them equal to the common
denominator.

The first denominator is missing two 2's, so multiply the first expression by 4/4.

The second denominator is missing a single 3, so multiply the second expression by 3/3.

Now we get the following:

EX: Simplify

The denominators x +3 and x -2 are already completely factored.
Therefore the common denominator must be (x + 3)(x - 2) .
Multiply the first term byand the second term by

	[Multiply to get each term.]
	[Simplify each numerator.]
	[Combine numerators.]
	[Simplify numerator.]
	[The numerator cannot be factored; no further simplification possible.]

EX: Simplify

First find the least common multiple of the denominators. It is necessary to factor them first.

List these factors in table form.

Factors of x^2 -1	Factors of x^2 + x	Least Common Multiple
(x + 1) , (x - 1)	x , (x + 1)

Move over the common factors, in this only (x + 1) .

Factors of x^2 -1	Factors of x^2 + x	Least Common Multiple
(x - 1)	x	(x + 1)

Move over the remaining factors from each term.

Factors of x^2 -1	Factors of x^2 + x	Least Common Multiple
	x	(x + 1) , (x - 1)

Factors of x^2 -1	Factors of x^2 + x	Least Common Multiple
		x , (x + 1) , (x - 1)

Therefore the least common multiple and common denominator is x(x +1)(x - 1) .
The first denominator is missing a single x, so multiply the first term by x/x.
The second denominator is missing (x - 1) , so multiply the second term by


	[Multiply out each term.]
	[Combine the numerators.]
	[The numerator can be factored.]

Since 2x^2 + x -1 is of the form Ax^2 + Bx + C , where A ≠ 0 , the middle term must be split.
The product of 2 and -1 is -2, and the sum is 1, so split using 2 and -1.

	[Group the terms by two.]
	[Factor each group.]
	[Reverse distribute.]
	[Now the numerator is completely factored.]

Replace the numerator with its factored form.

	[Now the (x + 1) term can be cancelled out.]
	[Now the rational expression is completely simplified.]

Subtracting Rational Expressions

Subtraction of rational expressions is done in the same way as addition. The only difference is that all signs in
the numerator of the second expression must be switched.

EX: Simplify

In this case, both expressions have the same denominator.
Combine the numerators, but switch the sign of each term in the second numerator.

Now just simplify the numerator

Both andare acceptable answers.

EX: Simplify

Find the common denominator. First factor the two denominators.

x^2 + 5x + 6 = (x + 2)(x + 3) and x^2 + 3x + 2 = (x +1)(x + 2)

The only common factor between the two denominators is (x + 2) , so the least common multiple and
common denominator is (x + 2)(x + 3)(x +1) .

The first denominator is missing (x + 1) , so multiply the first expression by

The second denominator is missing (x + 3) , so multiply the second expression by

	[Factor the denominators.]
	[Multiply by 1.]
	[Simplify.]
	[Combine numerators over common denominator.]
	[Change signs in second numerator.]
	[Combine like terms.]
	[Factor the numerator.]
	[Cancel out common factors.]

Some addition operations can turn into subtraction when the correct common denominator is found.

EX: Simplify

Notice that the denominators, x - 2 and 2 - x are -1 multiples of each other ( 2 - x = -(x - 2) ).

Because of this, we can change the second denominator to x -2 and change the operation to
subtraction.

	[Now there is a common denominator.]
	[Combine numerators over the common denominator.]
	[Remove parentheses and switch signs of the second numerator.]
	[Combine like terms.]
	[Factor the numerator.]
	[Cancel out common factors.]
	[Simplify.]