Using Rational Expressions
A Rational Expression is the ratio of two polynomials:
Using Rational Expressions
Using Rational Expressions is very similar to Using Rational Numbers (you may like to read that first).
Adding Rational Expressions
The easiest way to add Rational Expressions is to use the common denominator method (similar to adding fractions):
And then simplify the result.
Like in this example:
Example:
2x−2 + 3x+1 = 2 × (x+1) + (x−2) × 3(x−2) × (x+1)
(Comparing to the formula above: a is 2, b is x−2, c is 3, and d is x+1)
Then we simplify it:
= 2x+2 + 3x−6 x2+x−2x−2
= 5x−4 x2−x−2
Subtracting Rational Expressions
Subtracting is just like Adding:
Example:
2 x−2 − 3 x+1 = 2 × (x+1) − (x−2) × 3 (x−2) × (x+1)
And then simplify:
= 2x+2 − (3x−6) x2+x−2x−2
= −x + 8 x2−x−2
Multiplication
To multiply two Rational Expressions, just multiply the tops and bottoms separately, like this:
Example:
2x−2 × 3x+1 = 2×3(x−2)(x+1)
And then simplify:
= 6x2−x−2
Division
To divide two Rational Expressions, first flip the second expression over (make it a reciprocal) and then do a multiply like above:
Example:
First flip the second one over and make it a multiply:
2x−2 / 3x+1 = 2x−2 × x+13
Then do the multiply:
2 x−2 × x+13 = 2(x+1)3(x−2)
Simplifying
We should find any values that make the denominator zero, as these are not in the domain of the expression. Excluding these ensures the expression remains defined.
Example:
x2 − 1x + 1 is undefined when x = −1
Its Domain (the values that can go into the expression) does not include −1
Now, we can factor x2−1 into (x−1)(x+1) so we get:
(x−1)(x+1)(x+1)
It is now tempting to cancel (x+1) from top and bottom to produce:
x − 1
Its Domain now does include −1
But it is now a different function because it has a different Domain.