Type the whole expression in a single field, with parentheses around every numerator and denominator that has more than one term, and press “Calculate”. Below you will find a solved case, the changes that are allowed and the ones that are not. If you have an equation with an equals sign, use fraction equations; to learn how to find the excluded values by hand, read algebraic fractions and their domain.
Worked example
Simplify (x² − 1)/(x − 1) and say for which values of x the result is valid. In the calculator you type (x^2-1)/(x-1).
- Factor the numerator. x² − 1 is a difference of squares: (x − 1)(x + 1).
- Cancel the common factor. x − 1 appears on top and on the bottom, which leaves x + 1.
- Write down the restriction. The original denominator, x − 1, is zero at x = 1. That leaves the condition
x-1 ≠ 0.
Result: x + 1, with x ≠ 1.
Check: with x = 3, the fraction equals (9 − 1)/(3 − 1) = 4 and x + 1 also equals 4. At x = 1 the original fraction does not exist, even though x + 1 equals 2.
What cancels and what stays
You simplify an algebraic fraction by factoring the top and bottom and removing the factors that repeat. As with any fraction, it is in lowest terms when the only common factor is 1.
| Fraction | Result | Condition it writes | Why |
|---|---|---|---|
| (x² + 2x + 1)/(x + 1) | x + 1 | x+1 ≠ 0 | The numerator is (x + 1)²: one factor x + 1 cancels |
| (x³ − 8)/(x² − 4) | (x² + 2x + 4)/(x + 2) | x^2-4 ≠ 0 | They share x − 2; the restriction keeps the original denominator |
| (x² + 1)/(x + 1) | Unchanged | x+1 ≠ 0 | They have no factor in common |
| (x + 2)/(x + 3) | Unchanged | x+3 ≠ 0 | The x is a term of a sum, not a factor |
The first two rows simplify and the last two were already reduced. The conditions are written as polynomials, as in the third column; to see the excluded value, set the polynomial equal to zero (x + 1 = 0 gives x = −1).
Adding, subtracting, multiplying and dividing
Type the whole operation in the same field. The calculator finds the common denominator, reduces the result and gathers the conditions from every denominator that appears along the way. In a division, the values that make the divisor zero are excluded too. The calculator writes each condition as a polynomial not equal to zero, including the product of the denominators (x² − 1 ≠ 0), which repeats what the other two already say. The last column of the table shows the same information already solved.
| Operation | Result | Conditions it writes | Excluded values |
|---|---|---|---|
| 1/x + 1/(x + 1) | (2x + 1)/(x² + x) | x ≠ 0, x+1 ≠ 0, x^2+x ≠ 0 | x ≠ 0 and x ≠ −1 |
| 2/(x − 1) − 1/(x + 1) | (x + 3)/(x² − 1) | x-1 ≠ 0, x+1 ≠ 0, x^2-1 ≠ 0 | x ≠ 1 and x ≠ −1 |
| (x/(x + 1)) ÷ ((x − 1)/(x + 1)) | x/(x − 1) | x+1 ≠ 0, x-1 ≠ 0, x^2-1 ≠ 0 | x ≠ −1 and x ≠ 1 |
In the first row, with x = 2 the original sum is 1/2 + 1/3 = 5/6 and the result gives (2 · 2 + 1)/(4 + 2) = 5/6. Substituting like this works for checking any result. If the two values do not match, some step went wrong.
How to type the expression
Multiplication is always written with a sign, because with the variable x an implied product would be ambiguous. The symbols *, · and × all work, and parentheses and square brackets group terms.
| You type | Result |
|---|---|
(x^2+2*x+1)/(x+1) | Accepts, gives x + 1 |
x^-1+1 | Accepts, gives (x + 1)/x, with x ≠ 0 |
(0.5*x+1/3)/(x+2/3) | Accepts, gives 1/2, with x ≠ −2/3 |
(x^2+2x+1)/(x+1) | Rejects because the sign is missing in 2x; type 2*x |
(x+1)(x-1) | Rejects because the sign between parentheses is missing; type (x+1)*(x-1) |
x^9/x | Rejects because the numerator already exceeds degree 8 |
After a rejection, fix the sign or lower the degree and calculate again.
If your problem is a different one
This page works with expressions, not equations. If there is an equals sign and you are looking for x, fraction equations checks the solutions against these same restrictions. To split a fraction into simple terms, for example before integrating, there is partial fractions. When there is no variable and the nested quotient is numerical, complex fractions and fraction expressions are the place. To understand why a denominator cannot be zero, read zero denominator.
Sources
- Wikipedia, “Algebraic fraction”. It defines a fraction as being in lowest terms if the only factor common to the numerator and the denominator is 1; en.wikipedia.org/wiki/Algebraic_fraction, accessed October 17, 2026. The other figures on the page come from the calculator’s own engine.
Frequently asked questions
What is (x − 1)/(1 − x)?
It equals −1 for every x other than 1. The denominator is the opposite of the numerator, because 1 − x = −(x − 1), and the quotient of two opposites is −1. At x = 1 the original fraction is 0/0 and is undefined, so that restriction stays.
What does it mean when the result is 0 and there are still restrictions?
The expression equals 0 wherever it is defined, but not at the excluded points. For example, 1/x − 1/x reduces to 0 with x ≠ 0, because at x = 0 both original terms do not exist. The zero does not erase the exclusions.