Rational Algebraic Expressions Calculator

About the Rational Algebraic Expressions Calculator

A rational algebraic expression is a fraction whose numerator and denominator are polynomials, such as (x² − 9)/(x² + x − 6). Working with them follows the same rules as ordinary fractions: multiply across, flip and multiply to divide, and find a common denominator to add or subtract. The extra challenge is factoring polynomials and keeping track of the values of x where the expression is undefined.

This rational algebraic expressions calculator adds, subtracts, multiplies or divides two rational expressions in x. It combines them, finds and cancels the greatest common factor of the numerator and denominator with exact arithmetic, and lists the excluded values, the values of x that make an original denominator zero.

How to Use the Rational Algebraic Expressions Calculator

Enter the first numerator and first denominator as polynomials in x, using ^ for powers, such as x^2 - 9.

Choose the operation: add, subtract, multiply or divide.

Enter the second numerator and second denominator.

Coefficients can be whole numbers, decimals or fractions such as 1/2.

The Formulas

  add or subtract:  P1/Q1 ± P2/Q2 = (P1·(LCD/Q1) ± P2·(LCD/Q2)) / LCD
  multiply:         P1/Q1 × P2/Q2 = (P1·P2) / (Q1·Q2)
  divide:           P1/Q1 ÷ P2/Q2 = (P1·Q2) / (Q1·P2)
  simplify:         divide top and bottom by their greatest common divisor
  excluded values:  zeros of Q1 and Q2, and of P2 when dividing

Step-by-Step Example

Multiply (x² − 9)/(x² + x − 6) by (x − 2)/(x + 5).

  factor:   x² − 9 = (x − 3)(x + 3)       x² + x − 6 = (x + 3)(x − 2)
  multiply: (x − 3)(x + 3)(x − 2) / ((x + 3)(x − 2)(x + 5))
  cancel:   (x + 3)(x − 2) = x² + x − 6
  result:   (x − 3)/(x + 5)
  excluded: x ≠ −5, −3, 2

The cancelled factors still matter: at x = −3 and x = 2 the original product is undefined even though the simplified form has a value there.

Adding and Subtracting

To add or subtract, the fractions need a common denominator, ideally the least common denominator. If the denominators share a factor, the LCD uses it only once, which keeps the numbers smaller. For example, x/(x² − 4) − 2/(x² − 4) already share the denominator, so the difference is (x − 2)/(x² − 4). Since x² − 4 = (x − 2)(x + 2), the factor x − 2 cancels, leaving 1/(x + 2), with x ≠ 2 and x ≠ −2.

When the denominators share nothing, the LCD is their product. 2/(x − 3) + x/(x + 1) becomes (2(x + 1) + x(x − 3)) over (x − 3)(x + 1), which expands to (x² − x + 2)/(x² − 2x − 3). The numerator has no factor in common with the denominator, so that is the final answer.

Multiplying and Dividing

Multiplying is the most direct operation: multiply the numerators, multiply the denominators, then cancel. Factoring before you multiply makes the cancelling easy to see by hand. Dividing means multiplying by the reciprocal of the second expression. Because its numerator moves to the denominator, any value of x that makes the second numerator zero is also excluded: you cannot divide by zero.

How the Calculator Simplifies

Rather than relying on factoring, which only works when the factors are nice, the calculator finds the greatest common divisor of the numerator and denominator polynomials with the Euclidean algorithm, the same process used to reduce 12/18 to 2/3. All the arithmetic uses exact fractions, so there are no rounding errors. Finally it scales the result so the coefficients are whole numbers with no common factor and the denominator's leading coefficient is positive.

Excluded values are found as rational zeros of the denominators, using the rational root theorem. If a denominator has a factor such as x² + 1 or x² − 2 with no rational zeros, the calculator says so and names the factor, since its real zeros, if any, are excluded too.

Why Excluded Values Matter

Simplifying changes the formula but not the function's domain. The graph of (x² − 4)/(x − 2) looks exactly like the line x + 2, except for a hole at x = 2, where the original is undefined. When solving equations with rational expressions, any solution that is an excluded value must be rejected as extraneous, which is why listing them is part of every complete answer.

Checking Your Answer

Substitute a value of x that is not excluded into both the original problem and the simplified result; they must agree. For the example, at x = 0 the original gives (−9/−6) × (−2/5) = 1.5 × −0.4 = −0.6, and the simplified (x − 3)/(x + 5) gives −3/5 = −0.6. Testing two or three values catches most algebra slips, though only the full simplification proves equality.

Complex Fractions

A complex fraction has fractions inside its numerator or denominator, such as (1 + 1/x)/(1 − 1/x). One way to simplify it is to treat it as a division: rewrite the top as (x + 1)/x and the bottom as (x − 1)/x, then divide, which gives (x + 1)/(x − 1) with x ≠ 0 and x ≠ 1. Enter the two pieces with the divide operation to let the calculator do the cancelling, and remember that 0 stays excluded even though it no longer appears in the answer.

Understanding Your Result

The headline gives the simplified result.

The problem line restates the two expressions and the operation.

The combined line shows the expression before cancelling.

The cancelled line names the common factor removed.

The excluded values line lists the values of x that must be left out.

When Should You Use This Calculator?

Use it to check homework on adding, subtracting, multiplying and dividing rational expressions.

Use it to find excluded values before solving a rational equation.

Use it to simplify a complicated fraction of polynomials exactly.

Common Mistakes

Cancelling terms instead of factors. In (x + 3)/(x + 5), the x's do not cancel.

Adding numerators and denominators. 1/x + 1/y is not 2/(x + y).

Forgetting to flip when dividing. Multiply by the reciprocal of the second expression.

Dropping excluded values after cancelling. They still apply to the simplified form.