Algebraic Fractions — Simplify, Add & Divide

Algebraic fractions are fractions that contain variables in the numerator, denominator, or both. You handle them much like ordinary fractions, with one extra rule that governs every step: the work is valid only for values that keep the denominator nonzero. For example, $\frac{x+3}{x-2}$ is defined only when $x \ne 2$, and even if a later step cancels a factor, any value that made the original denominator zero stays excluded.

The core formulas and what the symbols mean

An algebraic fraction is also called a rational expression. The main difference from a numerical fraction is that common factors are often hidden until you factor, which is why factoring drives almost every operation here.

You can cancel a common factor, but you cannot cancel part of a sum. So $\frac{x+2}{x}$ does not simplify by "canceling the $x$" from only one term in the numerator.

Why the restrictions survive simplification

Here is the idea that explains the whole topic: restrictions come from the original expression, not the simplified one. When you factor and cancel, you are rewriting the expression into a shorter form, but the domain was already fixed by the starting denominator. Cancelling $(x-3)$ removes it from the picture, yet the value $x=3$ was already forbidden before you cancelled, so it stays forbidden. The simplified form is shorter; the domain is not.

To simplify an algebraic fraction:

1. Find any values that make the denominator zero.
2. Factor the numerator and denominator if possible.
3. Cancel only factors that appear in both the numerator and denominator.
4. Keep the original restrictions with the final answer.

For example,

Now the common factor $(x-3)$ can be canceled, so

but the original denominator shows that $x \ne 0$ and $x \ne 3$.

Worked example: simplify, then add

Simplify and add

Start by factoring the first fraction:

Now cancel the common factor $(x-1)$:

So the whole expression becomes

The denominators already match, so add the numerators:

The final result is

with the original restriction $x \ne 0$ and $x \ne 1$. The value $x=1$ is still excluded because it made the original denominator zero before simplification.

Division adds one move on top of this: multiply by the reciprocal. For example,

becomes

with restrictions $x \ne -1$ and $x \ne 0$. Here $x \ne 0$ matters twice: it keeps the denominator of $\frac{2}{x}$ nonzero, and it prevents the divisor from being undefined.

Try it, and the three errors to avoid

Simplify $\frac{x^2+5x+6}{x^2+2x}$ first, then add your result to $\frac{1}{x}$. As you work, watch the excluded values before and after simplification and confirm they do not change.

Three errors account for most lost marks:

Canceling terms instead of factors. You may cancel $(x-1)$ from $\frac{(x-1)(x+2)}{x(x-1)}$, but not the $x$ from $\frac{x+2}{x}$, because $x+2$ is a sum, not a single factor.

Forgetting excluded values. After simplifying, students often keep only the new denominator restriction. Restrictions come from the original expression.

Adding across different denominators too early. $\frac{1}{x} + \frac{1}{x+1}$ is not $\frac{2}{2x+1}$. You need a common denominator before you add the numerators.

Algebraic fractions appear throughout algebra because many formulas are ratios of expressions: simplifying rational expressions, solving equations, working with rates, and studying rational functions. The core habits stay the same: watch the denominator, factor early, and cancel only full factors.