Adding Algebraic Fractions
The sum of two or more algebraic fractions with the same denominator is itself an algebraic fraction. The numerator is obtained by adding the numerators, while the denominator remains unchanged: $$ \frac{A}{B} + \frac{C}{B} = \frac{A+C}{B} $$
How to add algebraic fractions
To add two or more algebraic fractions, proceed as follows:
- If the denominators are different, rewrite the fractions with a common denominator:
- Factor each denominator and identify the restrictions on the variables
- Find the least common multiple (LCM) of the denominators
- Use the invariant property of fractions to rewrite each term with the LCM as its denominator
- Add the numerators while keeping the common denominator.
Note. After carrying out the addition, it is often useful to factor the numerator as well, so the resulting fraction can be simplified to lowest terms.
A worked example
Consider the sum of the following algebraic fractions:
$$ \frac{x+1}{xy^2} + \frac{x-1}{x^2y} + \frac{y+x^2}{x^2y^2} $$
Here the denominators are not the same.
We therefore factor the denominators in order to determine the least common multiple of the polynomials:
$$ \text{lcm} (xy^2,x^2y,x^2y^2) = x^2y^2 $$
The least common multiple is x2y2.
Now apply the invariant property so that each fraction has this common denominator:
$$ \frac{x+1}{xy^2} \cdot \frac{x}{x} + \frac{x-1}{x^2y} \cdot \frac{y}{y} + \frac{y+x^2}{x^2y^2} $$
$$ \frac{x(x+1)}{x^2y^2} + \frac{y(x-1)}{x^2y^2} + \frac{y+x^2}{x^2y^2} $$
$$ \frac{x^2+x}{x^2y^2} + \frac{xy-y}{x^2y^2} + \frac{y+x^2}{x^2y^2} $$
At this point, all fractions share the same denominator.
We can therefore add the numerators directly:
$$ \frac{x^2+x+xy-y+y+x^2}{x^2y^2} $$
$$ \frac{2x^2+x+xy}{x^2y^2} $$
Next, factor the numerator:
$$ \frac{x(2x+y+1)}{x^2y^2} $$
And simplify:
$$ \frac{2x+y+1}{xy^2} $$
The final result is the sum of the original algebraic fractions, expressed in simplest form.
And the process can be repeated in the same way for more complex cases.
