Rational Expressions Calculator

One box, pre-filled with (x^2-9)/(x+3). Click Calculate. This rational expressions calculator reduces any rational expression you type to lowest terms, right on the page.

• Type your expression with ^ for powers and * for multiplication. The pre-filled (x^2-9)/(x+3) shows the exact format the rational expressions calculator expects.
• Or tap an example chip. Three chips prefill the other worked inputs of the rational expressions calculator, one tap each.
• Read the simplified form, rendered as a real math formula. The rational expressions calculator reports the answer in lowest terms, never as a decimal.

Two output shapes occur, and the rational expressions calculator handles both. Whole-division inputs like (x^2-9)/(x+3) collapse to the polynomial x − 3. Non-division inputs like (x^2+5x+6)/(x^2-4) stay fractional, reduced to (x+3)/(x−2). Try each chip and watch the rational expressions calculator switch shapes. Either way, the rational expressions calculator keeps full symbolic form.

This page is more than a static rational expressions calculator — it is also an AI tutor you can ask questions:

• Ask in plain words. “Simplify (x^2+8x+15)/(x^2-25) and list the excluded values.” The AI fills the box and runs this rational expressions calculator for you.
• Take a photo of your homework. The AI reads the expression and fills the rational expressions calculator form for you.
• Ask about the ideas, not just the answer. “Why can’t I cancel the x-squared terms?” The AI explains step by step. Every number still comes from the rational expressions calculator on this page, so nothing is guessed.
• Work backwards, then keep asking. Give the rational expressions calculator a simplified form and ask what original fraction produced it. Follow-up questions are welcome — the rational expressions calculator page keeps teaching.

A rational expression is a fraction whose numerator and denominator are both polynomials. The default (x^2-9)/(x+3) in the rational expressions calculator above is one. So is any polynomial-over-polynomial ratio the tool will face. Constants count too, because a constant is a zero-degree polynomial. Every fraction you have ever met is fair game for the rational expressions calculator. Some people search for a rational algebraic expressions calculator — longer name, same job.

One rule comes before everything: the denominator can never be zero. Before any canceling, the rational expressions calculator asks which x-values make the denominator vanish. For (x^2-9)/(x+3), set x + 3 = 0, and the answer is x = −3. So the expression is defined for all real numbers except x = −3, written x ≠ −3. Restrictions are not fine print; they are part of the rational expressions calculator answer itself. For (x^2+5x+6)/(x^2-4) the same step bans x = 2 and x = −2 — the rational expressions calculator lists both. Those banned inputs are the excluded values, and they travel with every answer the rational expressions calculator gives.

After canceling, (x^2-9)/(x+3) equals x − 3 — almost. The simplified form plots as the straight line y = x − 3. The original expression cannot produce a value at x = −3, so the line carries a hole at (−3, −6). One point is missing; everywhere else the two forms agree exactly. That hole is the picture behind every restriction the rational expressions calculator reports. Whenever this rational expressions calculator cancels a factor, picture the line and its missing point.

The method predates any software. Ray’s New Higher Algebra (1866) taught the rule the rational expressions calculator still runs today:

“Divide both terms of the fraction by any quantity that will exactly divide them… Or, Divide both terms by their greatest common divisor. Or, Resolve both terms into their prime factors, and then cancel those factors which are common. In algebraic fractions, the last is generally the best method.”

Modern algebra books keep that skeleton, and the rational expressions calculator runs it in three moves:

1. Find the excluded values first. Set the denominator equal to zero and solve; the rational expressions calculator bans those x-values before any canceling.
2. Factor the numerator and denominator completely — the step where most rational expressions calculator mistakes are born.
3. Cancel common factors — whole factors, never loose terms.

Canceling divides top and bottom by the same quantity, so the value never moves. That single fact powers every rational expressions calculator result on this page.

Start with the rational expressions calculator’s default input, (x²−9)/(x+3).

1. Excluded value: $x + 3 = 0$ gives $x = -3$, so $x \neq -3$ from the start.
2. Factor completely: $x^2 - 9$ is a difference of squares, so $x^2 - 9 = (x-3)(x+3)$.
3. Cancel the common factor $(x+3)$:

$\dfrac{x^2-9}{x+3} = \dfrac{(x-3)(x+3)}{x+3} = x-3$

The simplified form is $x - 3$. Type (x^2-9)/(x+3) into the rational expressions calculator above and it returns exactly that, rendered as a formula. The chip (2*x^2-8)/(x-2) collapses the same way — the rational expressions calculator reduces it to 2x + 4. This is the rational expressions calculator behavior the picture above drew.

Now the first example chip of the rational expressions calculator, (x²+5x+6)/(x²−4) — a non-division case.

1. Excluded values: $x^2 - 4 = 0$ gives $x = 2$ or $x = -2$.
2. Factor both polynomials: $x^2+5x+6 = (x+2)(x+3)$ and $x^2-4 = (x-2)(x+2)$.
3. Cancel the common factor $(x+2)$:

$\dfrac{x^2+5x+6}{x^2-4} = \dfrac{(x+2)(x+3)}{(x-2)(x+2)} = \dfrac{x+3}{x-2}$

Nothing else is common, so the answer stays a fraction: $\dfrac{x+3}{x-2}$, with $x \neq 2$ and $x \neq -2$. The rational expressions calculator chip (x^2-7x+12)/(x^2-16) ends the same way, at (x−3)/(x+4). Paste (x^2+5x+6)/(x^2-4) into the rational expressions calculator to confirm it.

Why is canceling legal at all? Ray’s 1866 proposition covers it:

“The value of a fraction is not altered, when both terms are multiplied or divided by the same quantity.”

Run the rational expressions calculator’s logic in reverse to build a rational expression:

1. Start from the simplified form $x - 3$.
2. Multiply by a form of 1: $\dfrac{x+3}{x+3}$, allowed because both terms scale by the same quantity.
3. Expand the numerator: $(x-3)(x+3) = x^2 - 9$, giving $\dfrac{x^2-9}{x+3}$ with the restriction $x \neq -3$ attached.

One step of the rational expressions calculator disassembles what this build assembled. Every messy rational expression is some tidy form dressed up in common factors. Ask the rational expressions calculator to undo your build whenever you want a check.

1. Canceling terms instead of factors. In $\dfrac{3x+6}{3}$, crossing out the 3s to leave $x + 6$ is wrong. Factor first: $3x+6 = 3(x+2)$, and $\dfrac{3(x+2)}{3} = x+2$. The rational expressions calculator always factors before touching anything.
2. Dropping the excluded values. Once (x^2-9)/(x+3) reduces to $x-3$, the restriction $x \neq -3$ still applies. Canceling hides a zero denominator; it does not erase it. The rational expressions calculator keeps the restriction visible.
3. Quitting before factoring is complete. In $\dfrac{4x+8}{x^2+5x+4}$, full factoring gives $\dfrac{4(x+2)}{(x+1)(x+4)}$, which proves nothing cancels. The rational expressions calculator never skips that check.