Polynomial Factoring Calculator

Factoring a polynomial writes it as a product of simpler polynomials, such as x² − 5x + 6 = (x − 2)(x − 3). This calculator takes out common factors, finds rational roots with the rational root theorem, splits the remaining quadratic when possible, and says when a factor cannot be split over the rationals. Each answer is expanded back to check.

Try:

How to use the polynomial factoring calculator

  1. Type the polynomial using x, such as 2x^3 - 3x^2 - 11x + 6.
  2. Press Calculate.
  3. Read the factored form, the roots and the graph.

Formula

f(r)=0  ⟺  (x−r) is a factor of f(x)f(r) = 0 \iff (x - r) \text{ is a factor of } f(x)

Worked example: x² − 5x + 6

  • Polynomial: x^2 - 5x + 6

✓ Answer checked

(x−3)(x−2)\left(x - 3\right)\left(x - 2\right)
Rational roots
3,  23,\; 2
123452468x = 3x = 2y = p(x)
Each linear factor (x − r) gives a place where the graph crosses or touches the x-axis.
Step-by-step working (2 steps)

  1. Write in standard form

    Highest power first.

    x2−5x+6x^{2} - 5x + 6

  2. Factor the quadratic

    Its discriminant is a perfect square, so it splits into two linear factors.

    (x−3)(x−2)\left(x - 3\right)\left(x - 2\right)
Formulas used
f(r)=0  ⟺  (x−r) is a factorf(r) = 0 \iff (x - r) \text{ is a factor}
How this was checked
  • ✓ Multiplying the factors back out gives exactly the original polynomial.

Frequently asked questions

How do I factor x² − 5x + 6?

Find two numbers that multiply to 6 and add to −5: −2 and −3. So x² − 5x + 6 = (x − 2)(x − 3).

What does 'irreducible over the rationals' mean?

The polynomial cannot be written as a product of factors with rational coefficients, like x² + 1 or x² − 2.

How do I factor a difference of squares?

a² − b² = (a − b)(a + b). For example, x⁴ − 16 = (x² − 4)(x² + 4) = (x − 2)(x + 2)(x² + 4).

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