Factoring Calculator (Polynomial & Number)
Prime factorization, divisors, GCD/LCM, and quadratic factoring with discriminant and roots.
MATHTwo-in-one factoring tool. Switch between Number mode (prime factorization, divisor list, GCD, LCM of two integers) and Quadratic mode (factor ax²+bx+c into a(x−r₁)(x−r₂) using the discriminant b²−4ac).
Number mode does trial division by primes up to √n, then enumerates every divisor and computes GCD via the Euclidean algorithm with LCM = a·b/gcd. Quadratic mode evaluates the discriminant D = b²−4ac: D > 0 gives two distinct real roots and the factored form a(x−r₁)(x−r₂); D = 0 gives a repeated root and a(x−r)²; D < 0 reports complex conjugate roots and notes the polynomial is irreducible over the reals. Worked example: 60 = 2² × 3 × 5, divisors {1,2,3,4,5,6,10,12,15,20,30,60}, GCD(60,24)=12, LCM=120. Quadratic x²−5x+6 has discriminant 1, roots 2 and 3, factored (x−2)(x−3).
Factoring Calculator (Polynomial & Number)
Factor positive integers into primes (with all divisors, GCD, and LCM) or factor a quadratic ax² + bx + c into a(x − r₁)(x − r₂). Shows the discriminant and handles complex-root cases.
How Factoring Works
Every integer greater than 1 has a unique prime factorization — this is the Fundamental Theorem of Arithmetic. Trial division by primes 2, 3, 5, 7, ... up to √n always terminates because any composite n must have a factor ≤ √n. Once you have the prime decomposition, every divisor of n is a product of subsets of those primes (with multiplicity), so the divisor list and the divisor count follow immediately. The number of divisors of n = p₁^a₁ · p₂^a₂ · ... is (a₁+1)(a₂+1)... — e.g. 60 = 2²·3·5 has (2+1)(1+1)(1+1) = 12 divisors.
The quadratic formula comes from completing the square. Starting from ax² + bx + c = 0, divide by a, move c/a across, and add (b/2a)² to both sides: (x + b/2a)² = (b² − 4ac) / 4a². Taking square roots gives x = (−b ± √(b² − 4ac)) / 2a. The expression under the radical, D = b² − 4ac, is the discriminant. It tells you everything about the roots before you compute them.
When D > 0 the polynomial has two distinct real roots and factors as a(x − r₁)(x − r₂). When D = 0 there is one repeated root and the factored form is a(x − r)². When D < 0 the roots are complex conjugates and the polynomial is irreducible over the reals — it factors only over ℂ. By the Rational Root Theorem, a quadratic with integer coefficients factors over the rationals exactly when D is a perfect square; otherwise the roots are irrational (still real, but the factored form contains surds).
Handles standard ax² + bx + c with real coefficients and positive-integer factoring. For higher-degree polynomials, symbolic factoring, or factoring over finite fields, use a CAS such as Wolfram Alpha or SymPy.