What is a rational expression?
A rational expression is a fraction with a polynomial on top and a polynomial on the bottom, like (x + 3)/(x − 2). It works the same way as a number fraction such as 6/8, except the pieces contain a variable.
Because the bottom can't be zero, every rational expression comes with a short list of banned values. In (x + 3)/(x − 2), x can be anything except 2. Those banned values are called excluded values, and they matter as much as the simplified answer.
To use the calculator, type the expression on one line with parentheses around each numerator and denominator, for example (x^2+5x+6)/(x^2-4). Without the parentheses, x^2+5x+6/x^2-4 means something quite different.
How do you simplify rational expressions step by step?
Factor, note the excluded values, then cancel common factors. Simplifying rational expressions follows the same idea as reducing 6/8 to 3/4, where you divide out the shared 2.
- Factor the numerator completely.
- Factor the denominator completely.
- Set each factor of the denominator equal to zero to find the excluded values.
- Cancel any factor that appears on both the top and the bottom.
- Write the simplified fraction together with the excluded values that are no longer visible.
The answer still says x ≠ −2 even though x + 2 has disappeared. The restriction x ≠ 2 isn't written separately because x − 2 is still sitting in the denominator, so anyone reading it can see it.
Why do excluded values stay after you cancel?
The simplified form has to equal the original expression for every allowed x, and the original was never defined at the excluded values. Cancelling changes how the expression looks, but it can't make a forbidden input allowed.
Take (x² − 1)/(x − 1). The top factors to (x + 1)(x − 1), so the calculator returns x + 1, x ≠ 1. If you plug in x = 1 to the original, you get 0/0, which has no value. The line y = x + 1 has a point at x = 1, but the graph of the original has a hole there.
Copy the restriction into your own work too. The answer x + 1 on its own describes a slightly different function, and the "x ≠ 1" part is what makes the two match.
Which factoring patterns do you need for rational expressions?
Most textbook problems use three patterns: a greatest common factor, a difference of squares, and a trinomial. If you can spot those, you can simplify almost any rational expression from an Algebra 1 or Algebra 2 course.
| Pattern | Example | Factored |
|---|---|---|
| Greatest common factor | 4x + 8 | 4(x + 2) |
| Difference of squares | x² − 4 | (x + 2)(x − 2) |
| Trinomial with a = 1 | x² + 5x + 6 | (x + 3)(x + 2) |
| Perfect square trinomial | x² + 6x + 9 | (x + 3)² |
| GCF, then difference of squares | 2x² − 8 | 2(x + 2)(x − 2) |
Always pull out a common factor first, then look for the other patterns. If a factoring step is the part you're stuck on, the factoring calculator shows that step by itself in more detail.
How do you add rational expressions with unlike denominators?
Rewrite both fractions over a common denominator, add the numerators, then factor and simplify. It's the same process as 1/3 + 1/4, where both fractions become twelfths.
Nothing cancels here, since 2x + 1 isn't a factor of the bottom. The calculator still checks for common factors before it stops.
Subtraction works the same way, but put the second numerator in parentheses so the minus sign reaches every term. For 3/x − 2/(x + 2), the top becomes 3(x + 2) − 2x = x + 6, and the result is (x + 6)/(x(x + 2)).
How do you multiply and divide rational expressions?
To multiply, factor everything and cancel across the whole product before you multiply out. To divide, flip the second fraction and multiply.
Try (x² − 4)/(x + 3) × (x + 3)/(x − 2). The top factors to (x + 2)(x − 2), so x + 3 and x − 2 both cancel. The rational expression simplifier returns x + 2, x ≠ −3, x ≠ 2, keeping both values that made an original denominator zero.
With division there's one extra restriction to watch. The value x = 1 is excluded because it makes x² − 1 zero inside the divisor, even though x − 1 ends up on top. To type the division, use / between the two bracketed fractions: (x/(x+1)) / (x^2/(x^2-1)).
Why can you cancel factors but not terms?
You can only cancel something that multiplies the whole numerator and the whole denominator. A term that is added or subtracted isn't a factor, so it can't be crossed out.
The classic slip is (x + 2)/2 = x. Test it with x = 4: the left side is 6/2 = 3, not 4. The 2 on top is added to x, so it isn't a factor of the numerator. The simplifying rational expressions calculator leaves it as (x + 2)/2, because nothing cancels.
The same trap shows up as (x² + 9)/(x + 3). It's tempting to cancel the 3 with the 9, or to treat x² + 9 like x² − 9. But x² + 9 doesn't factor over the real numbers, so nothing cancels. Factor first, and only cross out a whole bracket that appears on both sides of the fraction bar.