Factoring Calculator

Factor polynomials and integers step-by-step. Supports GCF, trinomials, difference of squares, and more.

Use ^ for powers (x^2), * for multiplication (3*x), and only the variable x.

How this factoring calculator works

This tool focuses on the most common algebra factoring tasks you see in high school and early college: factoring integers, quadratics, and many higher-degree polynomials with integer coefficients.

For polynomials in \(x\), the calculator tries these methods in order:

  1. Normalize the expression (remove spaces, combine like terms, sort by degree).
  2. Greatest common factor (GCF) of all coefficients.
  3. Difference of squares patterns like \(a^2 - b^2\).
  4. Quadratic trinomials \(ax^2 + bx + c\) using the AC method.
  5. Quadratic in disguise (e.g. \(x^4 - 5x^2 + 4\) by substituting \(u = x^2\)).

1. Factoring integers into primes

When you choose Integer mode, the calculator computes the prime factorization of the number.

Example:

\(360 = 2^3 \cdot 3^2 \cdot 5\)

Algorithm outline:

  • Take the absolute value \(n\).
  • Repeatedly divide by 2 while even.
  • Try odd divisors up to \(\sqrt{n}\).
  • If the remaining \(n > 1\), it is prime.

2. Factoring out the greatest common factor (GCF)

For a polynomial like \(6x^2 + 9x\), every term shares a common factor of 3:

\(6x^2 + 9x = 3(2x^2 + 3x)\)

The calculator:

  • Extracts the GCF of all coefficients.
  • Divides each term by that GCF.
  • Continues factoring the inside if possible.

3. Factoring trinomials \(ax^2 + bx + c\)

For quadratics, we use the AC method (also called factoring by grouping):

  1. Compute \(A = a \cdot c\).
  2. Find integers \(m, n\) such that \(m \cdot n = A\) and \(m + n = b\).
  3. Rewrite \(bx\) as \(mx + nx\).
  4. Factor by grouping.

Example: \(6x^2 + 11x - 35\)

\(a = 6,\; b = 11,\; c = -35,\; A = 6 \cdot (-35) = -210\)
We need \(m, n\) with \(m \cdot n = -210\) and \(m + n = 11\). One pair is \(21\) and \(-10\).
\(6x^2 + 11x - 35 = 6x^2 + 21x - 10x - 35\)
\(= 3x(2x + 7) - 5(2x + 7)\)
\(= (3x - 5)(2x + 7)\)

4. Difference of squares and special patterns

The calculator also recognizes common patterns such as:

  • \(a^2 - b^2 = (a - b)(a + b)\)
  • \(a^2 + 2ab + b^2 = (a + b)^2\)
  • \(a^2 - 2ab + b^2 = (a - b)^2\)

Example:

\(x^4 - 16 = (x^2)^2 - 4^2 = (x^2 - 4)(x^2 + 4) = (x - 2)(x + 2)(x^2 + 4)\)

5. Limitations

  • Only one variable \(x\) is supported.
  • Coefficients should be integers (no decimals or fractions).
  • Very high degrees or huge coefficients may not factor in reasonable time.
  • If no integer factorization exists, the tool reports the polynomial as irreducible over the integers.

Factoring practice examples

  • \(x^2 - 9 = (x - 3)(x + 3)\)
  • \(4x^2 - 12x = 4x(x - 3)\)
  • \(x^4 - 5x^2 + 4 = (x^2 - 1)(x^2 - 4) = (x - 1)(x + 1)(x - 2)(x + 2)\)

FAQ