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Algebra Calculator
Type an algebraic expression to simplify it, multiply out the brackets or factor it completely. Answers are exact (fractions and radicals, never rounded) and every result comes with the rules that produced it.
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Reads as
Result
Step by step
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How to use the Algebra Calculator
- Pick a mode: Simplify, Expand or Factor.
- Type your expression using ^ for powers, / for fractions and sqrt() for roots, or tap the symbol buttons. 2x and x(x+1) are read as multiplication.
- Check the preview to make sure the expression was read the way you meant it.
- Read the result and steps. The result updates as you type; press the green button or Enter to recalculate.
- Copy the answer as plain text or as LaTeX for documents.
Algebra rules the calculator applies
| Rule | Pattern | Example |
|---|---|---|
| Combine like terms | ax + bx = (a + b)x | 3x + 2x − 4 = 5x − 4 |
| Product of powers | xᵃ · xᵇ = xᵃ⁺ᵇ | x² · x⁵ = x⁷ |
| Power of a power | (xᵃ)ᵇ = xᵃᵇ | (2x³)² = 4x⁶ |
| Distributive law | a(b + c) = ab + ac | 2(x − 4) = 2x − 8 |
| Square of a binomial | (a ± b)² = a² ± 2ab + b² | (x + 3)² = x² + 6x + 9 |
| Difference of squares | a² − b² = (a − b)(a + b) | x² − 16 = (x − 4)(x + 4) |
| Sum / difference of cubes | a³ ± b³ = (a ± b)(a² ∓ ab + b²) | x³ − 8 = (x − 2)(x² + 2x + 4) |
| Simplify a radical | √(a²b) = a√b | √72 = 6√2 |
| Cancel a common factor | (ac)/(bc) = a/b, c ≠ 0 | (x² − 1)/(x − 1) = x + 1 |
| Pythagorean identity | sin²u + cos²u = 1 | sin²x + cos²x + x = x + 1 |
A reliable order for factoring
- Take out the greatest common factor first. 2x³ − 8x = 2x(x² − 4). Everything after this step is easier.
- Count the terms. Two terms: look for a difference of squares or a sum or difference of cubes. Three terms: try a perfect-square trinomial, then the ac method. Four terms: try grouping in pairs.
- For trinomials ax² + bx + c, find two integers that multiply to a·c and add to b. For 6x² + 11x − 10, a·c = −60 and the pair is 15 and −4, so 6x² + 15x − 4x − 10 = 3x(2x + 5) − 2(2x + 5) = (3x − 2)(2x + 5).
- For cubics and higher, test candidate roots ±p/q, where p divides the constant term and q divides the leading coefficient. Each root r gives a factor (x − r).
- Check every factor again – x⁴ − 16 becomes (x² − 4)(x² + 4) and then (x − 2)(x + 2)(x² + 4).
Typing tips
- Brackets matter:
1/2xmeans (1/2)·x, while1/(2x)puts 2x in the denominator. The "Reads as" preview shows the interpretation. - Write
x^(1/3)orcbrt(x)for cube roots and|x|orabs(x)for absolute value. - Functions such as
sin,cos,lnandlog(base 10) are treated as single quantities, so sin(x)² − sin(x) factors as sin(x)(sin(x) − 1).
Frequently asked questions
What is the difference between simplify, expand and factor?
Expanding multiplies out every bracket and combines like terms, giving a sum of terms such as x² + 5x + 6. Factoring does the reverse and writes the expression as a product, such as (x + 2)(x + 3). Simplifying chooses whichever correct form is shortest: it combines like terms, applies exponent rules and cancels common factors in fractions.
How do I type exponents, roots and fractions?
Use ^ for powers (x^3, 2^(n+1)), sqrt() for square roots, cbrt() or ^(1/3) for cube roots and / for division with brackets around longer numerators and denominators, for example (x^2-1)/(x+1). Multiplication can be written with * or left out: 3x, 2(x+1) and (x+1)(x-1) all work.
How do you simplify a rational expression?
Write everything over one denominator, factor the numerator and the denominator, then cancel the factors they share. For (x² − 9)/(x² + 6x + 9) the numerator is (x − 3)(x + 3) and the denominator is (x + 3)², so the expression simplifies to (x − 3)/(x + 3), valid for x ≠ −3.
Why does my polynomial not factor?
Some polynomials are irreducible over the rational numbers, which means they cannot be written as a product of lower-degree polynomials with rational coefficients. x² + 1 and x² − 2 are examples. When a quadratic has real but irrational roots, the calculator also shows the factorization over the real numbers using square roots.
Which factoring methods does it use?
It factors out the greatest common factor, recognises differences of squares, perfect-square trinomials and sums or differences of cubes, uses the ac method for trinomials, factors by grouping, finds rational roots with the rational root theorem and searches for higher-degree factors with integer coefficients.
Can it handle more than one variable?
Yes. Any single letters can be variables, so expressions such as 4x² − 9y², a² + 2ab + b² or xy + 3x + 2y + 6 can be expanded, simplified and factored. The letter e means Euler's number and pi means π.
Is the answer always correct?
Results are calculated with exact rational arithmetic and then checked by evaluating both your expression and the result at several random values. A green check mark means they agreed. Very large expressions may be refused because the engine runs inside your browser.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.
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