Factorization Calculator
Factor numbers into prime factors or polynomials into irreducible factors. Supports integers and quadratic expressions.
How to Use This Calculator:
- Select Prime Factorization for numbers or Polynomial Factorization for quadratic expressions
- Enter your value:
- For prime factorization: any integer greater than 1
- For polynomial factorization: quadratic in form ax²+bx+c (e.g., x^2-5x+6)
- Click "Factorize"
- View the step-by-step factorization process
- Copy the result if needed
Understanding Factorization
Prime Factorization
Breaking down a number into its prime components. Every integer greater than 1 is either prime or can be uniquely represented as a product of primes (Fundamental Theorem of Arithmetic).
Example: 60 = 2 × 2 × 3 × 5 = 22 × 3 × 5
Polynomial Factorization
Expressing a polynomial as a product of irreducible polynomials. For quadratics (degree 2), we find roots using the quadratic formula and express as (x - root₁)(x - root₂).
Example: x²-5x+6 = (x-2)(x-3)
Applications of Prime Factors
Used in cryptography (RSA), simplifying fractions, finding GCD/LCM, and solving Diophantine equations. Prime factorization is fundamental in number theory.
Applications of Polynomial Factors
Essential for solving equations, graphing polynomials, partial fractions in calculus, and simplifying algebraic expressions in engineering and physics problems.
Factorization Examples
Prime Factorization Examples
Polynomial Factorization Examples
Factorization Calculator FAQ
What's the difference between prime and polynomial factorization?
Prime factorization breaks numbers into prime number components, while polynomial factorization breaks algebraic expressions into simpler polynomial factors. Both reveal fundamental building blocks of their respective mathematical objects.
Why does prime factorization stop at 1?
By definition, 1 is neither prime nor composite. The prime factorization process stops when we reach 1 because we've fully decomposed the original number into primes.
Can all quadratic polynomials be factored?
Over real numbers, only quadratics with non-negative discriminant (b²-4ac ≥ 0) can be factored. Irreducible quadratics (with negative discriminant) require complex numbers for factorization.
How does the calculator handle fractions in polynomial roots?
When roots are fractions, the calculator attempts to display them as exact rational factors (e.g., (2x-1) for root x=½). For irrational roots, decimal approximations are shown.
What's the largest number this calculator can factor?
For prime factorization, the practical limit depends on server resources, but numbers up to 15-20 digits should work. Extremely large numbers may time out due to computational complexity.
Why Use Our Factorization Calculator?
Master Factorization Techniques
Our factorization calculator helps students, teachers, and professionals break down numbers and polynomials into their fundamental components. Whether you're solving math problems, preparing for exams, or working on real-world applications, this tool provides clear, step-by-step factorization results.