Mathematics
Factoring a polynomial step by step
Type a polynomial in x: get its factorisation with the steps, from the common factor to the roots found by synthetic division, the special products recognised, and the real and complex roots.
How to factor a polynomial
Factoring a polynomial means writing it as a product of polynomials of lower degree. The order of moves is always the same: first take out the factor common to every term, then look for special products, then find roots with the factor theorem and synthetic division, and finally deal with quadratics.
Synthetic division rests on the factor theorem: if a number a makes the polynomial zero, then (x − a) is a factor. The possible rational roots are fractions whose numerator divides the constant term and whose denominator divides the leading coefficient. For 2x⁴ − 2x³ − 14x² + 2x + 12, once 2 is taken out, the roots −2, −1, 1 and 3 give 2(x + 2)(x + 1)(x − 1)(x − 3).
When a quadratic ax² + bx + c is left, the discriminant Δ = b² − 4ac decides: if it is a perfect square the roots are rational; if positive but not a square the quadratic factors only over the reals, with irrational roots; if negative it is irreducible. Some polynomials with no roots still split into two quadratics, such as x⁴ + 4 = (x² − 2x + 2)(x² + 2x + 2).
Common mistakes
- Forgetting to take out the common factor first: it makes every later step harder.
- Swapping (x − a) and (x + a): if the root is 3, the factor is (x − 3).
- Calling x² + 1 irreducible over the complex numbers too: over the reals it is, over the complex numbers it is (x + i)(x − i).
Frequently asked questions
How do I find the roots to try?
Take the divisors of the constant term and divide them by the divisors of the leading coefficient, with both signs. For 6x² − x − 2: ±1, ±2, ±1/2, ±1/3, ±2/3, ±1/6. Here −1/2 and 2/3 work.
Can every polynomial be factored?
Over the complex numbers, always, into linear factors: that is the fundamental theorem of algebra. Over the reals, always into linear and quadratic factors. Over the rationals many polynomials, such as x² − 2, are irreducible.
What is the difference between a root and a factor?
Two sides of the same thing: a is a root of the polynomial exactly when (x − a) is a factor of it.
How this calculation works
Common factor: divide by the HCF of the coefficients. Factor theorem: P(a) = 0 ⇔ (x − a) divides P(x). Possible rational roots: p/q with p dividing the constant term and q dividing the leading coefficient. Quadratic ax² + bx + c: Δ = b² − 4ac, roots x = (−b ± √Δ) / 2a and ax² + bx + c = a(x − x₁)(x − x₂). Quartic: (a₁x² + b₁x + c₁)(a₂x² + b₂x + c₂) with integer coefficients, by comparing coefficients.
Related calculators
Complex numbers
Operations, rectangular, polar and exponential form, powers and n-th roots, with the Argand diagram.
Linear and quadratic equations
Roots, discriminant, vertex and factored form of ax² + bx + c = 0.
Differential equations
First and second-order linear differential equations with constant coefficients, with solution and graph.
Synthetic division
Divides a polynomial by (x − a) and factors it.