🧭 Polynomial Factoring Calculator
Factor quadratic expressions of the form ax² + bx + c into linear factors. Enter the coefficients to find the roots and the complete factored form.
What is this tool?
Factoring a polynomial means rewriting it as a product of simpler polynomials. For example, x² − 5x + 6 factors into (x − 2)(x − 3). Factoring is a core algebra skill that reveals the roots of an equation (where the polynomial equals zero), simplifies rational expressions, and unlocks techniques for solving inequalities and graphing functions.
The most commonly factored polynomial is the quadratic ax² + bx + c. If the discriminant b² − 4ac is a perfect square, the quadratic factors neatly into two linear terms with rational coefficients. If the discriminant is positive but not a perfect square, the factors involve irrational numbers. If the discriminant is negative, the quadratic does not factor over the real numbers.
This factoring calculator handles quadratic expressions. You enter the coefficients a, b and c, and it finds the roots using the quadratic formula, then writes the factored form as a·(x − r₁)(x − r₂) where r₁ and r₂ are the roots. If the roots are complex, it tells you the expression is irreducible over the reals. All computation runs locally in your browser.How it works
The calculator first computes the discriminant Δ = b² − 4ac. If Δ ≥ 0, it finds the roots r₁ = (−b + √Δ) / 2a and r₂ = (−b − √Δ) / 2a. The factored form is then a·(x − r₁)(x − r₂). When a = 1, the factored form is simply (x − r₁)(x − r₂).
For the special case where the discriminant is a perfect square, the roots are rational and the factored form has integer or fractional coefficients. For example, x² − 5x + 6 has discriminant 25 − 24 = 1, roots 3 and 2, and factors as (x − 3)(x − 2). When the discriminant is negative, the calculator reports that the polynomial is irreducible over the real numbers and displays the complex roots for reference.How to use
- Enter the coefficient a (must not be zero).
- Enter the coefficient b.
- Enter the coefficient c.
- Click Calculate to see the roots and factored form.
- Check whether the polynomial factors over the real numbers.
Frequently Asked Questions
When can a quadratic not be factored?
A quadratic ax² + bx + c cannot be factored into real linear factors when its discriminant b² − 4ac is negative. In this case, the roots are complex numbers. The polynomial is called irreducible over the reals. For example, x² + 1 has discriminant −4 and does not factor over the reals.
What is the difference between factoring and the quadratic formula?
They are two paths to the same destination. The quadratic formula gives you the roots directly; factoring rewrites the polynomial as a product of linear terms. Factoring is more useful when you need the factored form itself (for simplifying rational expressions or solving inequalities), while the quadratic formula is more universal and works even when factoring is difficult.
What if the leading coefficient a is not 1?
When a ≠ 1, the factored form includes a as a leading multiplier: a·(x − r₁)(x − r₂). For example, 2x² − 6x + 4 has roots 1 and 2, and factors as 2(x − 1)(x − 2). Some textbooks factor out the GCF first: 2x² − 6x + 4 = 2(x² − 3x + 2) = 2(x − 1)(x − 2).
Can this calculator factor higher-degree polynomials?
This calculator handles quadratics (degree 2). For cubics and higher, factoring requires more advanced techniques such as the rational root theorem, synthetic division or grouping. However, once you find one rational root r of a cubic, you can divide out (x − r) to get a quadratic, which you can then factor using this tool.
Tips & Advice
Before using the factoring calculator, try the AC method by hand: multiply a and c, then look for two numbers that multiply to give ac and add to give b. If such numbers exist, the quadratic factors over the integers. For example, for 6x² + 7x + 2, ac = 12 and you need two numbers that multiply to 12 and add to 7: those are 3 and 4. Rewrite 7x as 3x + 4x, factor by grouping and you get (2x + 1)(3x + 2). Always check your factored form by multiplying it back out: (x − 3)(x − 2) = x² − 5x + 6. If you get back the original polynomial, your factoring is correct. When solving inequalities, the factored form tells you the sign of the polynomial on each interval: the polynomial changes sign at each simple root, so test one point in each interval to determine where the polynomial is positive or negative.
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