Using Rational Expressions

A Rational Expression is the ratio of two polynomials:

Rational Expression

You can read more on our page on Rational Expressions

Using Rational Expressions

Using Rational Expressions is very similar to Using Rational Numbers (you might like to read that first).

Adding Rational Expressions

The easiest way to add Rational Expressions is to use the common denominator method:

common-factor-formula

Like in this example:

Example:

2   +   3   =   2·(x+1)  + (x-2)·3
x-2 x+1 (x-2)(x+1)

 

Which can then be simplified to:

  =   2x+2  + 3x-6    =   5x-4
x2+x-2x-2 x2-x-2

Subtracting Rational Expressions

Subtracting is just like Adding:

Example:

2   -   3   =   2·(x+1)  - (x-2)·3
x-2 x+1 (x-2)(x+1)

 

Which can then be simplified to:

  =   2x+2  - (3x-6)   =   -x+8
x2+x-2x-2 x2-x-2

Multiplication

To multiply two Rational Expressions, just multiply the tops and bottoms separately, like this:

Example:

2   ×   3   =   2·3
x-2 x+1 (x-2)(x+1)

 

Which can then be simplified to:

  =   6
x2-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:

2   /  3   =   2   ×   x+1
x-2 x+1 x-2 3

 

Then do the multiply:

2   ×   x+1   =   2(x+1)
x-2 3 3(x-2)

 

Simplifying

When simplifying a rational function be careful to respect where the lower polynomial is equal to zero

Example:

4x is undefined where 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:

4x

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.

 

 
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