Polynomial Factoring Calculator
Factor a quadratic polynomial ax² + bx + c instantly. Enter the coefficients below to find the factors, roots, discriminant, calculation steps, and verification.
Factor a Quadratic Polynomial
Enter the coefficients a, b, and c for ax² + bx + c.
What Is Polynomial Factoring?
Polynomial factoring is the process of rewriting a polynomial as a product of simpler expressions. Factoring can make equations easier to solve, simplify algebraic expressions, and reveal the roots of a polynomial.
For a quadratic polynomial, the general form is:
where a ≠ 0. When the quadratic has real roots r₁ and r₂, its factored form is:
This calculator uses the discriminant and the quadratic formula to determine the roots and construct the factored form.
The Discriminant
The discriminant is one of the most useful tools for understanding a quadratic polynomial.
- Δ > 0: two distinct real roots.
- Δ = 0: one repeated real root.
- Δ < 0: no real roots; the roots are complex.
If the discriminant is non-negative, the quadratic can be represented using real linear factors.
How to Factor a Quadratic Polynomial
Consider a quadratic polynomial:
The roots can be found with the quadratic formula:
Suppose the two roots are r₁ and r₂. The polynomial can then be expressed as:
For example, consider:
The two numbers that multiply to 6 and add to −5 are −2 and −3. Therefore:
Expanding the factors confirms the original polynomial:
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Frequently Asked Questions
What is a factored polynomial?
A factored polynomial is written as a product of simpler expressions. For example, x² − 5x + 6 can be factored as (x − 2)(x − 3).
How do you factor ax² + bx + c?
One method is to find the roots using the quadratic formula. If the roots are r₁ and r₂, the factored form is a(x − r₁)(x − r₂).
What happens when the discriminant is zero?
When Δ = 0, the quadratic has one repeated root and can be written as a squared factor, such as (x − r)², multiplied by the leading coefficient when necessary.
Can a quadratic with a negative discriminant be factored?
It cannot be factored into real linear factors because it has no real roots. It can, however, be factored over the complex numbers.
Is polynomial factoring the same as solving a quadratic?
They are closely related but not identical. Factoring rewrites the polynomial as a product, while solving finds the values of x that make the polynomial equal to zero. The factors can be used directly to identify those roots.