Polynomial Factoring Calculator
Factor any polynomial up to degree 3 and instantly find its roots and factored form. Ideal for students solving quadratics or cubics in algebra class or checking homework.
About this calculator
Factoring a polynomial means rewriting it as a product of simpler expressions. For a quadratic ax² + bx + c, the key tool is the discriminant: Δ = b² − 4ac. If Δ > 0, there are two distinct real roots; if Δ = 0, one repeated root; if Δ < 0, two complex roots. The roots are found via the quadratic formula: x = (−b ± √Δ) / (2a). Once roots r₁ and r₂ are known, the factored form is a(x − r₁)(x − r₂). For degree-3 polynomials, rational root theorem and synthetic division are applied first to reduce the cubic to a quadratic, which is then factored using the same discriminant approach.
How to use
Suppose you want to factor 2x² + 5x − 3. Enter degree = 2, coeff_a = 2, coeff_b = 5, coeff_c = −3. The discriminant is: Δ = 5² − 4(2)(−3) = 25 + 24 = 49. Since √49 = 7, the roots are x = (−5 + 7) / 4 = 0.5 and x = (−5 − 7) / 4 = −3. The fully factored form is therefore 2(x − 0.5)(x + 3), which simplifies to (2x − 1)(x + 3).
Frequently asked questions
How do I factor a polynomial with a negative discriminant?
A negative discriminant (b² − 4ac < 0) means the quadratic has no real roots — its factors involve complex numbers. The roots take the form x = (−b ± i√|Δ|) / (2a), where i is the imaginary unit. While these roots are not visible on a real-number graph, the factored form is still valid over the complex numbers. In practical algebra courses, such polynomials are often called irreducible over the reals.
What factoring methods are used for degree-3 polynomials?
Cubic polynomials are typically factored by first applying the rational root theorem to find a candidate integer root, then using synthetic division to reduce the cubic to a quadratic. The resulting quadratic is factored using the discriminant and quadratic formula. Special patterns like the sum or difference of cubes (a³ ± b³) also apply in certain cases. This calculator handles all these steps automatically.
When should I use a polynomial factoring calculator instead of doing it by hand?
Manual factoring is straightforward for simple trinomials with small integer coefficients, but becomes error-prone for cubics or quadratics with large or fractional coefficients. A calculator is especially useful when you need to verify your work, handle non-integer roots, or quickly identify whether a polynomial is irreducible over the reals. It's also valuable during exam review when speed and accuracy both matter.