2026-09-29 9 min

Polynomial Factorization Calculator

Factor quadratics ax²+bx+c over integers with discriminant, rational roots and GCF steps. Free online factoring calculator.

Calculator

Quadratic Coefficients

a — x² coefficient Help

Zero → linear case

b — x coefficient Help

Middle term

c — constant Help

Constant term

Results update live as you type

Status:x² − 5x + 6(x − 3)(x − 2)

Factorization Summary

Input1x² + -5x + 6

Δ / GCF1 / 1

Factored(x − 3)(x − 2)

Roots3, 2

Results

Optimal

Factored Form

(x − 3)(x − 2)

Product of linear factors where possible

Discriminant Δ

b² − 4ac decides the root type

Roots

3, 2

Zeros of the quadratic

GCF of Terms

Pull out before splitting

Factor Type

Two rational factors

Classification from the discriminant

Roots come from the quadratic formula in double precision and each factorization is verifiable by substitution and FOIL.

Visualizations

|a| vs |b| vs |c| vs |Δ|

1|a|

5|b|

6|c|

1|Δ|

Factorability

Splits ✓100%

|Root₁| vs |Root₂|

|r₁|3

Formula

The discriminant decides whether integer factors exist; the quadratic formula produces the roots that become the factors.

ax² + bx + c = a(x − r₁)(x − r₂), with x = (−b ± √Δ)/2a

Alternative Form

Δ = b² − 4ac; GCF pulled out first

Variables

Symbol	Name	Unit	Description
a, b, c	Quadratic coefficients	numbers	Terms of ax² + bx + c
Δ	Discriminant	number	b² − 4ac; verdict on root type
r₁, r₂	Roots	numbers	Zeros that become (x − r) factors
GCF	Common factor	integer	Shared divisor extracted before splitting

Explanation

The parabola sketch shows opening direction and discriminant verdict; integer splits appear as clean factor pairs.

Derivation

ax²+bx+c = 0 → complete the square → x = (−b±√(b²−4ac))/2a. Integer roots need Δ a perfect square (rational-root structure).

Step-by-Step Calculation

Follow each step to understand how the result is calculated from your inputs.

Step 1

Pull out the GCF of the integer coefficients first.

GCF(|a|, |b|, |c|)

gcf:1

Result: GCF is 1 — nothing to pull out

Step 2

Compute the discriminant.

Δ = b² − 4ac

delta:-5² − 4(1)(6) = 1

Intermediate: Δ = 1

Result: Δ = 1

Step 3

Classify from the discriminant sign and square status.

Δ > 0 / = 0 / < 0

type:two-rational

Result: Two rational roots — splits over integers

Step 4

Apply the quadratic formula for the roots.

x = (−b ± √Δ) / 2a

r1:3

r2:2

Result: x = 3 or x = 2

Step 5

Write the factored form from the roots.

a(x − r₁)(x − r₂)

factored:(x − 3)(x − 2)

Result: (x − 3)(x − 2)

Step 6

Verify by substituting a root back.

a·r² + b·r + c = 0?

check:0

Result: Check ≈ 0 ✓ (x² − 5x + 6)

Final Answer

(x − 3)(x − 2)

Worked Examples

See how the calculator works with different scenarios.

Scenario

Classic integer split.

Inputs

a:1

b:-5

c:6

Calculation

Δ = 25−24 = 1; roots 2, 3

Result

(x − 2)(x − 3)

Interpretation

Two integer factors; FOIL back to check.

Understanding Your Result

Your Result: Factored: (x − 3)(x − 2)

Typical range: Δ > 0 splits (rational if perfect square); Δ = 0 squares; Δ < 0 irreducible over reals.

What This Means

Factored form lists the exact linear pieces whose product rebuilds the quadratic — and whose zeros solve it.

Why It Matters

Factoring solves equations, simplifies rational expressions, finds break-even points and prepares partial fractions.

Professional Interpretation

Always extract the GCF first — hidden common factors mask perfect squares and inflate root arithmetic.

Recommendations

•Pull the GCF out first
•Read Δ before hunting factors
•Verify roots by substitution
•FOIL the factors back
•Use the formula when Δ is not square

Educational Content

Introduction

Factoring reverses FOIL: it compresses a quadratic into the linear pieces that generate it.

Concept Explanation

Roots become factors via a(x − r₁)(x − r₂); the discriminant predicts whether those pieces stay rational.

Applications

▸Solving quadratic equations

▸Break-even and projectile zeros

▸Simplifying rational expressions

▸Partial-fraction setup

▸Graphing x-intercepts

▸Integer-puzzle factor hunts

Benefits

•GCF head-start shown
•Discriminant verdict explained
•Roots computed exactly
•Factored form emitted
•Verification built in

Limitations

•Quadratics (degree 2) only
•Integer/rational splits emphasized
•Cubics need other methods

Best Practices

•Extract GCF before splitting
•Compute Δ before guessing
•List factor pairs of a·c
•Substitute roots to verify
•Expand back as final proof

Common Misconceptions

Myth: Every quadratic factors over integers.

Truth: Only perfect-square Δ gives rational roots; x² + x + 1 never splits over integers.

Myth: Δ < 0 means a mistake.

Truth: It means no real roots — a valid, common outcome (e.g. x² + 1).

Myth: GCF extraction is optional.

Truth: Skipping it hides squares: 2x²+4x+2 splits only after pulling out 2.

Myth: Double roots count as one factor.

Truth: They give a squared factor (x − r)² — multiplicity 2 matters.

Practical Tips

•Extract the GCF before anything else
•Compute Δ before hunting pairs
•Perfect-square Δ means clean split
•Test p/q candidates from a and c
•Substitute each root to verify
•FOIL factors back as proof
•Treat Δ = 0 as a squared factor

Common Mistakes

Skipping the GCF extraction

Divide all terms by GCF(|a|,|b|,|c|) first, then split the cofactor.

Guessing factors when Δ is not square

Non-square Δ means irrational roots — use the formula, not integer pairs.

Sign errors in (x − r) factors

Root r = 2 gives factor (x − 2); negate carefully for negative roots.

Forgetting the leading a outside

Keep a as a multiplier: 2x²+7x+3 = (2x+1)(x+3), not (x+…)(x+…).

Missing the double-root square

Δ = 0 yields (x − r)², not a single (x − r).

Assumptions

This calculator makes the following assumptions to provide estimates:

Quadratic intent: a ≠ 0 for the quadratic path; a = 0 is reported linear.

Real coefficients: a, b, c are real numbers.

Integer GCF: GCF step applies to integer-valued triples.

Exact arithmetic: Inputs are exact values.

Display rounding: Roots show up to 6 decimals.

Limitations

Be aware of these limitations when using this calculator:

Degree 2 only: Cubics and higher need rational-root/synthetic pages.

Rational emphasis: Irrational roots are reported, not factored.

Real scope: Complex factors are noted, not expanded.

Tolerance edge: Near-zero discriminants snap to double-root verdict.

Frequently Asked Questions

Answers to common questions about this calculator.

What does factoring a quadratic mean?

Writing ax² + bx + c as a product of linear factors, e.g. x² − 5x + 6 = (x − 2)(x − 3). Factored form exposes the roots directly.

How do I use this calculator?

Enter a, b and c for ax² + bx + c. You get the discriminant, the GCF of the terms, the roots, and the factored form (or the reason it will not split over integers).

What is the discriminant?

Δ = b² − 4ac. Positive perfect square → two rational roots; positive non-square → real but irrational; zero → one double root; negative → no real roots.

How does the GCF step help?

Pull shared factors out first: 2x² + 4x + 2 = 2(x² + 2x + 1) = 2(x + 1)². Factoring the GCF first makes the remaining split far easier.

What is the rational-root check?

Any rational root p/q has p dividing c and q dividing a. This calculator tests that structure: integer-friendly roots produce clean (x − r₁)(x − r₂) factors.

What is a double root?

When Δ = 0 both roots coincide: x² + 6x + 9 = (x + 3)². The parabola just touches the x-axis at its vertex.

When is a quadratic irreducible?

Over the reals when Δ < 0 (e.g. x² + 1), and over the integers when Δ is not a perfect square (e.g. x² + x + 1 has irrational-free but non-integer roots only via formula).

How do I verify a factorization?

Expand it back with FOIL — this page shows the discriminant path so you can cross-check each root by substitution: a·r² + b·r + c must equal 0.

What if a = 0?

Then it is not quadratic but linear bx + c, with the single root x = −c/b. The calculator reports that degenerate case explicitly.