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Addition Of Algebraic Fractions - how to add fractions with algebra

Updated on August 15, 2012


Adding algebraic fractions can be done in exactly the same way as adding normal fractions with numbers. So all you need to do is make a common denominator – that is make the values on the bottom of both fractions the same. Just be careful with expanding brackets and simplifying the expressions. So the main steps to follow are:

Step A – Make common denominators.

Step B – Add the fractions together.

Step C – Simplify the numerator and factorise the numerator again (if possible) as you might be able to cancel out a factor with the denominator.

Example 1

Simplify:

2/(x+4) + 3/(x-2)

To make common denominators you can multiply the numerator and denominator of the first fraction by (x-2) and multiply the numerator and denominator of the second fraction by (x+4):

2(x-2)/(x+4)(x-2) + 3(x+4)/(x-2)(x+4)

The denominators of both fractions are now the same so you can now add the fractions:

[2(x-2) + 3(x+4)]/[(x-2)(x+4)]

The next step is to expand the brackets on the numerator and simplify the numerator. You don’t have to expand the brackets on the denominator unless the question wants the answer written a different way:

[2x-4 + 3x + 12]/[(x-2)(x+4)]

= [5x +8]/[(x-2)(x+4)]

Sometimes you can factorise the numerator and cancel one of the factors with the denominator, but in this example you cannot, so this is your final answer.

Let’s take a look at one more example:

Example 2

Simplify:

(3x-2)/(x-1)² + 5/(x-1)

To make common denominators you can multiply the numerator and denominator of the second fraction by (x-1). The first fraction can be left as it is:

(3x-2)/(x-1)² + 5(x-1)/(x-1)²

The denominators of both fractions are now the same so you can now add the fractions:

[(3x-2) + 5(x-1)]/(x-1)²

The next step is to expand the brackets on the numerator and simplify the numerator. You don’t have to expand the brackets on the denominator unless the question wants the answer written a different way:

[3x-2 + 5x-5)]/(x-1)²

= (8x – 7)/(x-1)²

The numerator of this fraction cannot be factorised, so nothing else will cancel off. This will be your final answer.

Subtracting algebraic fractions can be done in a similar to adding algebraic fractions. All you need to is to make common denominators and then subtract the fractions. However, you will have to take extra care with the negative sign in the middle of the algebraic fractions. If you would like to see some examples on the subtraction of algebraic fractions then make sure you read the next page.

Subtracting Algebraic Fractions. A simple guide on how to take away fractions with algebra.

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