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Adding and Subtracting Rational Expressions Calculator

ANSWER
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Fill in both fractions, choose + or −, then press =.
Over the LCD
top
bottom
top
bottom
The two bottoms are factored and the LCD is built from them — each factor at the highest power it appears in either, which is usually smaller than multiplying them together.
How these rational expressions combine, step by step
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Simplify Rational Expressions Calculator

Multiplying the two denominators together always works, and that is why so many answers to these questions are correct but unfinished — the fraction comes out bigger than it needed to be, and has to be reduced all over again.

This adding and subtracting rational expressions calculator with steps builds the LCD instead, and puts the product beside it so you can see the difference. It handles adding and subtracting rational algebraic expressions of any size the four boxes will take.

Adding and Subtracting Rational Expressions Calculator logo — a free tool from Monkza

What It Works Out, and Why

Give it 1/(x²−4) + 1/(x+2) and it factors both bottoms first: (x−2)(x+2) and (x+2). Multiplying those gives x³ + 2x² − 4x − 8. The LCD is only (x−2)(x+2), because (x+2) sits in both and is taken once rather than twice. Both are on screen, the larger one struck through.

The restrictions are written down at that point, before anything else happens — here x ≠ 2 and x ≠ −2, taken from the original denominators rather than from the LCD or from whatever the answer reduces to. That ordering is the whole reason they come out right.

Then the numerators are combined over the LCD and the result is reduced if it still shares a factor. With like denominators it goes straight to the numerators; with unlike denominators it does the rewriting for you. And when you are subtracting it prints the second numerator again with every sign changed, because by then it is rarely one term and the minus reaches all of it.

The arithmetic is exact whole numbers throughout, so nothing is a rounded decimal and the working never disagrees with the answer.

How to Use This Adding and Subtracting Rational Expressions Calculator

Choose add or subtract, fill in the four boxes — top and bottom of each fraction — and press =. The line under the buttons shows how your input was read, so you can check it as you type rather than after.

The answer appears with its restrictions beneath it. One tap opens the LCD working, and under that the full step by step: factor, restrictions, build the LCD, the sign flip if you are subtracting, combine, reduce. If you only need that last part, the simplify rational expressions calculator does it alone.

Frequently Asked Questions

How do you add rational expressions with unlike denominators?
Factor both denominators, build the LCD by taking each distinct factor at the highest power it reaches in either, rewrite both fractions over it, then add the numerators. The denominator does not change once you are over the LCD.
What is the LCD of two rational expressions?
The least common denominator — every factor of both denominators, each at its highest power. It is usually smaller than multiplying the denominators together, which also works but leaves a bigger fraction to reduce afterwards.
How do you subtract rational expressions?
Exactly as you add them, except the minus applies to the whole second numerator once it is over the LCD. That numerator is usually more than one term by then, and changing only its first sign is the commonest mistake in the topic.
What are the restrictions, and do they change?
Every value that makes either original denominator zero, and no, they do not change. They are fixed by the denominators you started with, whatever the answer ends up looking like.

Factor first, build the LCD rather than multiplying, take the restrictions from the original bottoms, and watch the minus. Those four cover the topic.