Polynomial Factor Calculator: Find and Check Factors

A polynomial factor calculator rewrites a polynomial as a product of simpler factors, such as turning x² − 5x + 6 into (x − 2)(x − 3). Enter the polynomial’s coefficients in descending powers; the calculator returns its factors and, when supported, any real or complex roots.

Formula

This calculator handles a quadratic polynomial of the form ax² + bx + c, where a is the coefficient of x², b is the coefficient of x, and c is the constant term. For a nonzero a, the quadratic formula gives the roots:

r₁ = (−b + √(b² − 4ac)) / (2a)
r₂ = (−b − √(b² − 4ac)) / (2a)

The corresponding factorization is a(x − r₁)(x − r₂). The calculator’s first two outputs are r₁ and r₂; its third output is the discriminant Δ = b² − 4ac. All inputs and outputs are numbers, so the unit is not applicable. The formula also works when the discriminant is negative only if the calculator supports complex-number arithmetic; the square-root expression here assumes a real-valued calculation.

Worked Example

Using the default coefficients a = 1, b = −5, and c = 6:

1. Find the discriminant: Δ = (−5)² − 4(1)(6) = 25 − 24 = 1.
2. Calculate the first root: r₁ = (−(−5) + √1) / (2 × 1) = (5 + 1) / 2 = 3.
3. Calculate the second root: r₂ = (−(−5) − √1) / (2 × 1) = (5 − 1) / 2 = 2.
4. Write the factors: x − 3 and x − 2. Thus x² − 5x + 6 = (x − 3)(x − 2).

The calculator reports roots 3.00 and 2.00, with discriminant 1.00. Expanding the factors confirms the result: (x − 3)(x − 2) = x² − 5x + 6.

What Polynomial Factoring Means

Factoring reverses polynomial multiplication. A polynomial is expressed as a product of lower-degree expressions, called factors. For example, x² − 9 factors as (x − 3)(x + 3). Factoring is useful for solving equations, simplifying rational expressions, and identifying where a polynomial equals zero.

The roots and factors are linked: if r is a root, then x − r is a factor. This is the factor theorem. In the example above, 2 and 3 make the quadratic equal zero, so its factors are x − 2 and x − 3. The leading coefficient matters: 2x² − 10x + 12 factors as 2(x − 2)(x − 3), not just (x − 2)(x − 3).

How to Use a Polynomial Factor Calculator

This page’s calculator uses three coefficients and therefore applies specifically to quadratics. Put the coefficient of x² in the first field, the coefficient of x in the second, and the constant term in the third. Include zero for a missing term. For instance, x² − 9 has a = 1, b = 0, and c = −9.

• Write the polynomial in descending powers before entering values.
• Keep negative signs with their coefficients; for x² + 4x − 5, enter 1, 4, and −5.
• Check that a is not zero. If a = 0, the expression is linear or constant, not a quadratic.
• Use the roots to construct factors, then expand the result to verify the coefficients.

Reading the Discriminant and Results

The discriminant Δ indicates the type of roots a quadratic has. If Δ is positive, there are two distinct real roots. If Δ is zero, there is one repeated real root. If Δ is negative, the roots are a complex conjugate pair rather than real numbers. A calculator limited to real arithmetic may show an error for a negative discriminant; that does not mean the polynomial cannot be factored over the complex numbers.

For example, x² + 1 has Δ = −4 and roots i and −i, where i² = −1. Over the real numbers it does not split into linear factors, while over the complex numbers it is (x − i)(x + i). Always check which number system a calculator uses and whether it displays exact values or rounded decimals.

Other Factoring Methods and Calculator Limits

Not every polynomial can be handled with the quadratic formula. A cubic, such as x³ − 6x² + 11x − 6, may factor by testing rational roots or using a dedicated higher-degree solver. Common hand methods include taking out a greatest common factor, grouping terms, recognizing a difference of squares, and using special-product identities.

Even a quadratic calculator gives roots rather than a complete symbolic explanation of how a polynomial was factored. It may also round roots, making a factorization built from displayed decimals only approximate. For exact integer or fractional factors, retain exact roots where possible and verify by multiplication. A polynomial with irrational roots can still factor over the real numbers, but its factors may contain radicals.

Frequently Asked Questions

How do I factor a quadratic polynomial?
Write it as ax² + bx + c, find its roots with the quadratic formula or another method, and use a(x − r₁)(x − r₂). Expand the factors to check your work.

What do I enter when a term is missing?
Enter zero for that coefficient. For x² − 9, the x-term is missing, so use a = 1, b = 0, and c = −9.

Can every polynomial be factored?
Every nonconstant polynomial has roots and linear factors over the complex numbers, but it may not factor into lower-degree polynomials with integer or real coefficients. The meaning of “factor” depends on the number system allowed.

What does a negative discriminant mean?
It means the quadratic has no real roots and has two complex conjugate roots. A real-only calculator may not display those roots, even though complex factors exist.

How can I check a factorization?
Multiply the proposed factors and compare the result with the original polynomial, including every coefficient and the constant term.

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in a | Coefficient of x^2 | - | 1 | 
in b | Coefficient of x | - | -5 | 
in c | Constant term | - | 6 | 
out factor1 | First factor root r1 | - | (-b + sqrt(b^2 - 4*a*c)) / (2*a) | 2
out factor2 | Second factor root r2 | - | (-b - sqrt(b^2 - 4*a*c)) / (2*a) | 2
out check | Discriminant | - | b^2 - 4*a*c | 2