Rational Expression

Simplifying Rational Expressions
A rational expression has a numerator and a denominator . Sometimes these are polynomials of the
form ax² + bx + c, or sometimes difference of squares (ax)² − b². The strategy for solving such
ex pressions involves factoring and then canceling :

consider:

First, factor the numerator: x² − 3x − 10 = (x − 5)(x + 3)

Then factor the denominator: 25 − x² = 5² − x² = (5 + x)(5 − x)

the result is

now we just have to cancel: (note that (5 − x) = −(x − 5))

Multiply and Divide Rational Expressions

Multiplying or dividing rational expressions involves first simplifying and canceling, as before:

First, change the operation from division to multiplication by taking the reciprocal as shown
be low :

Now just factor and cancel as before:

Least Common Multiple (LCM) of Polynomials
First Review the GCF: Recall that the Greatest Common Factor, or GCF, of two expressions is the
largest expression that divides evenly into both expressions:
example: the GCF of 8x²y⁴ and 12x³y² is 4x²y², because 4x²y² is the largest expression that divides
into both 8x²y⁴ and 12x³y² without a remainder. We use the GCF when we want to factor an expression
such as 8x²y⁴ + 12x³y².
The GCF of 8 and 12 is found by factoring 8 and 12 into its prime factors:

the GCF is 2 · 2 = 4 since both 8 and 12 have 2 · 2 or 4.
Now the least Common multiple, of LCM, of two or more expressions is found in a similar manner.
Let’s find the LCM of 8 and 12:

We start by pairing common factors, and then also including the factors that are unique to each.
Thus, the LCM of 8 and 12 is 2 · 2 · 2 · 3 = 24.
So we use the GCF when factoring. Also, we use the LCM when adding or subtracting Rational
Expressions:

Simplify

The LCM of 8x²y⁴ and 12x³y² must first be found. First, as we’ve found above, the LCM of 8 and
12 is 24. Now find the LCM of x² and x³. This is the opposite of finding the GCF. The GCF of
x² and x³ would be x². However, the LCM is x³, the variable with the higher numbered exponent .
Similarly, the LCM of y⁴ and y² is y⁴. Thus, the LCM of the two denominators would be 24x³y⁴
So the expression can be added as follows: