Subtracting Algebraic Fractions
The difference of two algebraic fractions with the same denominator is itself an algebraic fraction. The numerator is simply the difference of the two numerators, while the denominator remains unchanged. $$ \frac{A}{B} - \frac{C}{B} = \frac{A-C}{B} $$
How to subtract algebraic fractions
To work out the difference between two algebraic fractions:
- If the denominators are different, first rewrite the fractions with a common denominator.
- Factor each denominator and note the restrictions on the variable
- Find the least common multiple (LCM) of the denominators
- Use the invariant property of fractions so that each fraction has the LCM as its denominator
- Subtract the numerators, keeping the common denominator the same.
Note. After subtracting, it’s good practice to factor the numerator and reduce the fraction to its simplest form. The steps are essentially the same as when adding algebraic fractions.
A Worked Example
Consider the subtraction:
$$ \frac{x+1}{xy^2} - \frac{x-1}{x^2y} - \frac{y+x^2}{x^2y^2} $$
Here the denominators are all different.
So, we first factor the denominators and compute their least common multiple (LCM):
$$ \text{lcm}(xy^2,\;x^2y,\;x^2y^2) = x^2y^2 $$
The least common multiple of the denominators is x2y2.
Now apply the invariant property so each fraction is rewritten with this 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 three fractions share the same denominator.
So we subtract the numerators while keeping that denominator fixed:
$$ \frac{(x^2+x) - (xy-y) - (y+x^2)}{x^2y^2} $$
$$ \frac{x^2+x-xy+y-y-x^2}{x^2y^2} $$
$$ \frac{x-xy}{x^2y^2} $$
Now factor the numerator:
$$ \frac{x(1-y)}{x^2y^2} $$
And simplify:
$$ \frac{1-y}{xy^2} $$
The result is the algebraic difference in fully simplified form.
And that’s it.
