Polynomial Factorization Calculator - Factor Polynomials
Factor quadratic polynomials (ax² + bx + c) instantly. Shows step-by-step factorization using GCF, AC method, and grouping with complete verification.
Polynomial Inputs
Results
What is Polynomial Factorization?
Polynomial Factorization is the process of expressing a polynomial as a product of simpler polynomials. For quadratics in the form ax² + bx + c, this means finding factors (px + q)(rx + s) that multiply to give the original polynomial.
This calculator handles:
- GCF Extraction - Identifies and factors out the greatest common factor.
- Quadratic Factoring - Uses AC method and grouping for complete factorization.
- Prime Detection - Determines if a polynomial cannot be factored further.
To solve quadratic equations using the quadratic formula, visit our Quadratic Equation Solver for complete solutions.
For completing the square method, check our Completing the Square Calculator to transform quadratics.
To factor integers into prime factors, use our Prime Factorization Calculator for number theory.
Factorization Methods
The calculator uses these proven methods:
The process is verified by expanding the factored form to ensure it matches the original polynomial.
Polynomial Types
Factorable
Can be expressed as a product of linear factors with integer coefficients.
Prime
Cannot be factored further over the integers; irreducible.
How to Use This Calculator
Enter Coefficients
Input values for a, b, and c.
Review Polynomial
Check the displayed polynomial format.
Get Factored Form
See complete factorization and steps.
Benefits of Using This Calculator
- • Complete Method: Shows GCF extraction, AC method, and verification.
- • Prime Detection: Identifies when polynomials cannot be factored.
- • Verified Results: Expands factored form to confirm accuracy.
Factoring Techniques
1. Simple Trinomials (a = 1)
Find two numbers that multiply to c and add to b. Example: x² + 5x + 6 = (x + 2)(x + 3)
2. Complex Trinomials (a ≠ 1)
Use AC method: Find two numbers that multiply to ac and add to b, then factor by grouping.
Frequently Asked Questions (FAQ)
Q: What is polynomial factorization?
A: Polynomial factorization is the process of expressing a polynomial as a product of simpler polynomials. For quadratics, this means writing ax² + bx + c as (px + q)(rx + s).
Q: How do I know if a polynomial can be factored?
A: A quadratic polynomial can be factored over the integers if its discriminant (b² - 4ac) is a perfect square. If not, the polynomial may be prime or require factoring over real/complex numbers.
Q: What is the AC method for factoring?
A: The AC method finds two numbers that multiply to ac (the product of the leading coefficient and constant) and add to b (the middle coefficient). These numbers are used to split the middle term and factor by grouping.
Q: What is a prime polynomial?
A: A prime or irreducible polynomial is one that cannot be factored into polynomials of lower degree with integer coefficients. It's similar to a prime number in arithmetic.
Q: Should I always check for a GCF first?
A: Yes, always check for a greatest common factor (GCF) first. Factor out the GCF before attempting other factoring methods, as this simplifies the remaining polynomial.