Rational Expression Calculator

How to use this calculator

1. Enter coefficients, not the expressionFor x² − 4 enter a = 1, b = 0, c = −4. A missing term means that coefficient is zero.
2. Use a linear expression by setting a = 0A zero leading coefficient makes the expression linear, which the calculator handles as a single factor.
3. Read the holes and asymptotes separatelyThis is the point of the tool. Both are excluded from the domain, but they behave completely differently on a graph and exams test the distinction.
4. Second expression is for multiply and divide onlyIt is ignored in simplify mode. In divide mode the second fraction is inverted automatically.

Why the simplified form has a different domain

Cancelling a common factor changes the expression in a way that is easy to miss, and it is the single most tested idea in this topic.

Take (x² − 4) ÷ (x² − x − 6). Factor both: (x − 2)(x + 2) on top, (x − 3)(x + 2) underneath. The (x + 2) cancels, leaving (x − 2) ÷ (x − 3).

Those two expressions are not identical functions. The original is undefined at x = −2, because the denominator was zero there before cancelling. The simplified version is perfectly happy at x = −2, giving −4 ÷ −5 = 0.8.

So the correct answer is (x − 2) ÷ (x − 3), where x ≠ 3 and x ≠ −2. The second restriction has to be carried across explicitly, because nothing in the simplified expression shows it any more.

Graphically the difference is visible. At x = 3 the denominator survives cancellation, so the function runs off to infinity — a vertical asymptote. At x = −2 the factor cancelled, so the curve is perfectly well behaved either side and simply has a single point missing. That is a hole, or removable discontinuity.

The word removable is precise: the discontinuity can be removed by defining the function to equal 0.8 at that one point. An asymptote cannot be removed by any definition, because the function genuinely has no finite limit there.

Cancelling without recording the restriction is the standard way to lose marks on this topic, and it is why the breakdown lists holes and asymptotes as separate rows.

Worked example: (x² − 4) ÷ (x² − x − 6)

Factor the numerator. x² − 4 is a difference of two squares: (x − 2)(x + 2). Its roots are 2 and −2.

Factor the denominator. x² − x − 6 needs two numbers multiplying to −6 and adding to −1: those are −3 and 2, giving (x − 3)(x + 2). Its roots are 3 and −2.

Before cancelling anything, write down the restrictions from the denominator: x ≠ 3 and x ≠ −2.

Now cancel. The factor (x + 2) appears in both, so it goes.

Simplified: (x − 2) ÷ (x − 3).

The restrictions stay as they were: x ≠ 3 and x ≠ −2.

Of those two, x = 3 is a vertical asymptote — the factor survived in the denominator, so the function diverges there. And x = −2 is a hole, because the factor cancelled.

You can find the height of the hole by substituting into the simplified form: (−2 − 2) ÷ (−2 − 3) = −4 ÷ −5 = 0.8. So the graph passes through (−2, 0.8) except that this exact point is missing.

A quick sanity check on any cancellation: pick a value that is not excluded, say x = 0. The original gives (0 − 4) ÷ (0 − 0 − 6) = −4 ÷ −6 = 0.667. The simplified gives (0 − 2) ÷ (0 − 3) = −2 ÷ −3 = 0.667. They agree, as they must everywhere except at the hole.

Holes against asymptotes

The last two rows matter when a factor appears more than once. Compare the powers: whichever side has the higher power determines the behaviour. (x − 1)² in the numerator over (x − 1) in the denominator cancels down to (x − 1), giving a hole. The reverse gives an asymptote. Cancelling one occurrence at a time, as this calculator does, handles both correctly.

Multiplying, dividing and the operations that need more care

Multiplying rational expressions is the easy case. Factor everything, cancel across both fractions in any combination, and multiply what remains. There is no need to find a common denominator, and cancelling before multiplying is far less work than multiplying out first and factoring afterwards.

Dividing is multiplication by the reciprocal. Invert the second fraction and proceed as for multiplication. The one trap is the domain: the numerator of the second fraction becomes a denominator after inverting, so its roots become excluded values too. An expression divided by something that could be zero carries that restriction even though it looks like a numerator in the original.

Adding and subtracting are genuinely harder, and this calculator does not attempt them. They require a common denominator, which means factoring both denominators, finding the least common multiple of the factors, rewriting each fraction over it, and only then combining numerators. The result usually needs factoring again to see whether it simplifies. It is several steps of work where multiplication is one.

A specific warning about cancelling: only factors cancel, never terms. In (x + 3) ÷ (x + 5) the x values do not cancel, because they are terms added to something, not factors multiplying something. This is probably the most common error in all of algebra, and it produces answers that are confidently and completely wrong.

The test is simple. If the thing you want to cancel is joined to its neighbours by multiplication, it can go. If it is joined by addition or subtraction, it cannot.

Finally, complex fractions — a fraction whose numerator or denominator is itself a fraction — are handled by multiplying through by the least common denominator of all the small fractions, which clears them in one step.

Assumptions and limitations

• Handles quadratic numerators and denominators — cubics and higher must be factored by hand first
• Does not add or subtract rational expressions, which need a common denominator
• Quadratics with no real roots cannot be cancelled and are reported as already in lowest terms
• Roots are computed numerically, so irrational factors are shown as decimals rather than surds
• Does not identify horizontal or slant asymptotes, only vertical ones
• Does not compute the coordinates of a hole automatically — substitute into the simplified form
• Assumes a single variable x throughout

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