Factoring Calculator

Factors polynomials over the rational numbers and shows each step, such as a common factor, a special product or the rational root theorem. It also expands products and does synthetic division. Use it to check factoring homework.

How to use

1. On the "Factor" tab, type a polynomial into "Polynomial", for example x^2 - 5x + 6. You can leave out the multiplication sign.
2. Read the "Factored form" and the "Steps". Press "Copy result (LaTeX)" to copy the result.
3. To multiply out brackets, enter an expression such as (2x - 1)(x + 3) in "Expression to expand" on the "Expand" tab.
4. To divide by x − a, enter the "Dividend" and the "Divisor or number a" on the "Synthetic division" tab to get the "Quotient" and "Remainder".

How it works

• Polynomials in one variable are factored completely over the rationals with common factors, special products, quadratics, the rational root theorem and synthetic division. The degree can be up to 40.
• With two or more variables, only set forms are handled: common factors, x² − y², a² ± 2ab + b², a³ ± b³, quadratics in one letter and similar.
• Every result is expanded again and compared with the original expression automatically.
• Approximate real roots of factors that do not factor further are shown for reference.

Questions and answers

How do I factor x^2 - 5x + 6?

Find two numbers whose product is 6 and whose sum is −5: −2 and −3. The result is (x − 2)(x − 3).

What are the common factoring formulas?

a² − b² = (a − b)(a + b), a² ± 2ab + b² = (a ± b)², a³ + b³ = (a + b)(a² − ab + b²) and a³ − b³ = (a − b)(a² + ab + b²).

How does synthetic division work?

It divides by x − a using only a and the coefficients. Dividing x³ − 6x² + 11x − 6 by x − 1 gives the quotient x² − 5x + 6 and remainder 0.

Why does it say the polynomial does not factor further?

It cannot be split into factors with rational coefficients. For example, x² − 2 has the roots ±√2, so it does not factor over the rationals.