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Factoring Calculator

List the divisors and prime factors of any integer, or factor a quadratic into its roots.

Factor an integer

Factor a quadratic (ax² + bx + c)

How to use factoring calculator

  1. To factor an integer, type it into the "Integer to factor" field. The prime factorization and the full list of divisors appear as you type.
  2. To factor a quadratic, enter the coefficients a, b and c of ax² + bx + c in the three boxes below it.
  3. Read the factored form, the explanation and the roots in the result area — both panels recalculate on every keystroke.
  4. Clear a field to hide its result; non-integers and numbers above 1 trillion are flagged with a message.

Examples

Factor the integer 360

360 = 2³ × 3² × 5, and its 24 divisors run from 1 to 360.

Factor x² + 5x + 6

Two numbers that multiply to 6 and add to 5 give (x + 2)(x + 3), with roots −2 and −3.

Factor 2x² − 7x + 3

The discriminant is 25, so the roots are 3 and ½: 2(x − 3)(x − 0.5).

What is Factoring?

Factoring means breaking a mathematical object down into the pieces that multiply together to produce it — but the word covers two different jobs. Factoring an integer means finding its divisors and its prime factorization: expressing the number as a product of prime numbers, like 12 = 2² × 3. That prime factorization is unique for every integer (the fundamental theorem of arithmetic), and the full list of divisors follows directly from it.

Factoring a quadratic means rewriting ax² + bx + c as a product of two linear expressions, like x² + 5x + 6 = (x + 2)(x + 3). Factoring and roots are two views of the same thing: the expression equals zero exactly when x equals one of its roots. When a clean factorization is not obvious, the quadratic formula recovers the roots from the coefficients, and the discriminant b² − 4ac tells you what to expect before you start — positive means two real roots, zero means one repeated root, and negative means no real roots at all.

This calculator does both kinds of factoring instantly as you type. Enter any integer to see its divisors and prime factorization written with exponents, or enter the three coefficients of a quadratic to see its factored form, an explanation and its roots. Everything runs in your browser — no server, no upload, and the answers update with every keystroke.

Frequently asked questions

What if the discriminant is negative?

A negative discriminant (b² − 4ac < 0) means the quadratic has no real roots: its parabola never crosses the x-axis. It still has two complex-conjugate roots, but it cannot be factored using real numbers, and the calculator reports "no real factors".

Can the roots be fractions or decimals?

Yes. Roots come out as whole numbers only when the discriminant is a perfect square. Otherwise the quadratic formula produces fractions or irrational decimals, and the factored form is shown with decimal roots — for example 2(x − 3)(x − 0.5) instead of something with neat integers.

Is 1 a prime number?

No. A prime has exactly two distinct divisors — 1 and itself — while 1 has only one divisor. That is why 1 has no prime factorization: an empty product equals 1 by definition, which is why the calculator notes that 1 is not prime.

What is the largest integer I can factor?

Up to 1 trillion (1,000,000,000,000) in absolute value, negatives included. Larger numbers are rejected because trial division and the divisor list become slow and impractical to display long before the math stops working.

Why does the factorization use exponents like 2³?

It is shorthand for repeated multiplication: 2³ means 2 × 2 × 2. Exponents keep long factorizations readable — 1,000,000,000,000 is written 2¹² × 5¹² rather than a product of twenty-four individual numbers.