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Factoring Calculator

Free factoring calculator: factor any polynomial expression completely. Factor trinomials, binomials, and higher-degree

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How the Factoring Calculator Works

The factoring calculator gets 27,100 monthly searches — spiking to 49,500 in September and October (fall semester algebra) and again in April–May (spring finals). Related terms: "factor polynomial calculator" and "polynomial factoring calculator" together add another 7,200 searches; "factor trinomial calculator" adds 2,400 more. This is core algebra curriculum, and students rely on it to check homework and learn the method.

The calculator decomposes polynomial expressions by applying techniques in order: GCF extraction (greatest common factor), difference of squares, perfect square trinomial, AC method for trinomials, sum/difference of cubes, and grouping for higher-degree polynomials. For x2 + 5x + 6: find two numbers multiplying to 6 and adding to 5: 2 and 3. Factor: (x + 2)(x + 3). Verify by FOIL: x2 + 3x + 2x + 6 = x2 + 5x + 6 ✓.

Key Factoring Identities

  • Difference of squares: a2 - b2 = (a+b)(a-b)
  • Perfect square trinomial: a2 + 2ab + b2 = (a+b)2; a2 - 2ab + b2 = (a-b)2
  • Sum of cubes: a3 + b3 = (a+b)(a2-ab+b2)
  • Difference of cubes: a3 - b3 = (a-b)(a2+ab+b2)

Factoring Techniques: GCF, Difference of Squares, and AC Method

The factor polynomial calculator applies these techniques in sequence:

  1. GCF extraction: Always check first. For 6x2 + 9x, GCF = 3x. Factor: 3x(2x + 3).
  2. Difference of squares: a2 − b2 = (a+b)(a−b). For x2 − 49 = (x+7)(x−7). For 4x2 − 25 = (2x+5)(2x−5).
  3. Perfect square trinomial: a2 ± 2ab + b2 = (a±b)2. For x2 + 6x + 9 = (x+3)2.
  4. AC method for trinomials: For ax2 + bx + c, find two numbers multiplying to ac and adding to b. For 2x2 + 7x + 3: ac = 6, need numbers adding to 7: 6 and 1. Rewrite: 2x2 + 6x + x + 3 = 2x(x+3) + 1(x+3) = (2x+1)(x+3).
  5. Sum/difference of cubes: a3 + b3 = (a+b)(a2−ab+b2). For x3 + 8 = (x+2)(x2−2x+4).

Quadratic Factoring Calculator: Using the Discriminant

The quadratic factoring calculator (1,600 monthly searches) quickly checks factorability. For ax2 + bx + c: the discriminant Δ = b2 − 4ac tells you:

  • Δ = perfect square: Factors over integers. Example: x2 + 5x + 6, Δ = 25−24 = 1 (perfect square) → (x+2)(x+3).
  • Δ > 0, not perfect square: Factors over reals (irrational factors), not integers. Example: x2 − 3, Δ = 9 → (x+√3)(x−√3).
  • Δ < 0: No real factors — only complex. Example: x2 + x + 1, Δ = −3 → complex roots, can't factor over reals.
  • Δ = 0: Perfect square trinomial. Example: x2 + 4x + 4, Δ = 0 → (x+2)2.

Frequently Asked Questions

How do you know when a polynomial can't be factored?

A quadratic ax2+bx+c cannot be factored over integers if its discriminant (b2-4ac) is not a perfect square. If b2-4ac is negative, it has no real factors at all. For example, x2+x+1 has discriminant 1-4=-3 — not factorable over real numbers (only over complex numbers). A polynomial with no rational roots also can't be factored over rationals.