Algebraic Expression Simplifier

What does the expression simplifier do?

It rewrites the algebraic expression you type in its simplest form and shows how it got there. For polynomials it expands the brackets, distributes the products and combines like terms. For expressions with a variable in the denominator it writes everything as a single fraction, factors the numerator and the denominator and cancels the common factor. You can also give values to the letters and get the value of the expression.

Rules used

• Distribution: a(b + c) = ab + ac. When two brackets are multiplied, every term of the first is multiplied by every term of the second.
• Like terms: terms with the same letters and powers have their coefficients added. 3x and 5x combine, 3x and 5x² do not.
• Rational expressions: factor the numerator and the denominator and cancel a factor that appears in both. Terms that are added cannot be cancelled, only factors can.

Example

Type (x+2)(x-3) + 2x(x-1). Expanding gives x² − 3x + 2x − 6 + 2x² − 2x. The x² terms add up to 3x² and the x terms to −3x, so the result is 3x² − 3x − 6. A rational example: in (x^2 - 5x + 6)/(x^2 - 4) the numerator is (x − 2)(x − 3) and the denominator is (x − 2)(x + 2). The common factor (x − 2) cancels and (x − 3)/(x + 2) is left.

Excluded values

Where a cancelled factor is zero, the original expression is undefined. (x² − 1)/(x − 1) simplifies to x + 1, but at x = 1 the original denominator is zero. The tool lists these values under the result. Apart from them the simplified form equals the original expression.

Evaluating

Type values such as x = 2, y = -1 in the values box. The tool substitutes them into the original expression. When the values are whole numbers or fractions the result is given exactly as a fraction, with its decimal value next to it.

Limits

Intermediate steps are shown for polynomials and rational expressions. Fractions in several letters are reduced only when the numerator and the denominator can be factored: common factors, difference of squares, perfect squares and similar. For expressions with square roots, trigonometry, logarithms or π only the basic rules are applied (powers of the same base are merged, like terms are combined) and no intermediate steps are shown. Trigonometric identities are not applied. Every result is checked against the original expression with random numbers.