CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 2. Polynomials and Rational Functions

numerator and denominator may be canceled. The reason they can be “canceled” is that any expression divided by
itself is equal to 1. An identical expression in the numerator and denominator is just an expression being divided by
itself, and so equals 1.

• To multiply rational expressions, you should write the product of all the numerator factors over the product of
  all the denominator factors and then cancel identical factors.


• To divide rational expressions, you should rewrite the division problem as a multiplication problem. Multiply
  the first rational expression by the reciprocal of the second rational expression. Follow the steps above for
  multiplying.

To add or subtract rational expressions, it is essential to first find a common denominator. While any common
denominator will work, using the least common denominator is a means of keeping the number of additional
factors under control. Look at each rational expression you are working with and identify your desired common
denominator. Multiply each expression by an appropriate clever form of 1 and then you should have your common
denominator.
In both multiplication and division problems answers are most commonly left entirely factored to demonstrate
everything has been reduced appropriately. In addition and subtraction problems the numerator must be multi-
plied, combined and then re-factored. Example B shows you how to finish an addition and subtraction problem
appropriately.
Example A
Simplify the following rational expression.

x^2 + 7 x+ 12

x^2 + 4 x+ 3 ·

x^2 + 9 x+ 8

2 x^2 − 128 ÷

x+ 4

x− 8 ·

14

1

Solution: First factor everything. Second, turn division into multiplication (only one term). Third, cancel
appropriately which will leave the answer.

=((xx++^33 )()(xx++ 14 ))· 2 ((xx++^88 )()(xx+−^18 ))·((xx−+ 48 ))·^141

=(((

(x+ 3 )((x(+( 4 ()

((x(+(^3 )((x(+(^1 ()

·(((

(x+ 8 )((x(+(( 1 )

(^2) ((x(+( 8 )((x(−( 8 ()·
(x−^8 )
(x+^4 )·

14

1

=^142

= 7

In this example, the strike through is shown. You should use this technique to match up factors in the numerator
and the denominator when simplifying.
Example B
Combine the following rational expressions.

x^2 − 9

x^4 − 81 −

4 x

x^2 + 9 ÷

x− 3

2

Solution: