Separate the factoring domain before choosing a method
Factoring rewrites an object as a product, but the allowed factors depend on the object and number system. Integers decompose into primes. Polynomials decompose into lower-degree polynomials whose coefficients may be restricted to integers, rationals, reals, or complex numbers.
This calculator handles exact integer factorization and integer-coefficient quadratics over the integers or rationals. Integer mode uses one whole number; quadratic mode uses a, b, and c in ax^2 + bx + c with nonzero a.
- Identify the input type.
- Retain every sign and coefficient.
- Name the tested factor domain.
- Keep the original for verification.
Build a unique prime-power factorization
For every integer n with |n| greater than 1, the Fundamental Theorem of Arithmetic guarantees one prime factorization apart from factor order. Group repeated primes with exponents to create a compact prime-power form.
For example, 360 = 2^3 x 3^2 x 5. A negative input adds the unit -1 before the positive primes. Omitting it would make the product check return the wrong integer.
Formula notes
Integer factorization: |n| = p1^e1 x p2^e2 x ... x pk^ek
- Factor the absolute value.
- Record prime multiplicities.
- Restore -1 when negative.
- Multiply back to n.
Handle zero, units, primes, and composites precisely
Zero has no unique finite prime factorization because every nonzero integer divides it. One and negative one are units, neither prime nor composite. Beginning the primes at 2 preserves uniqueness.
A positive integer greater than 1 is prime when its only positive divisors are 1 and itself; otherwise it is composite. For a negative integer, apply that classification to its absolute value and carry the sign as -1.
- 0: no finite prime factorization.
- 1 and -1: units.
- Prime: two positive divisors.
- Composite: more than two.
Derive divisor properties from exponents
If n = p1^e1 x ... x pk^ek, every divisor chooses an exponent from 0 through ei. Therefore divisor count is the product of ei + 1, while divisor sum uses one geometric series per prime.
Euler's totient counts positive values up to n that are coprime to n. The radical multiplies distinct primes once. Even exponents identify squares; exponents no greater than one identify square-free values.
Formula notes
Positive divisor count: tau(n) = product of (ei + 1)Positive divisor sum: sigma(n) = product of (pi^(ei + 1) - 1) / (pi - 1)Euler totient: phi(n) = n x product of (1 - 1/pi)
- Tau counts divisors.
- Sigma sums them.
- Phi counts coprime values.
- Radical keeps distinct primes.
Extract polynomial content before testing a quadratic
For ax^2 + bx + c, first extract the coefficient GCF. Include a negative sign when needed so the primitive polynomial has a positive leading coefficient. The remaining coefficients are then coprime as a group.
For 12x^2 + 10x + 2, the content is 2 and the primitive expression is 6x^2 + 5x + 1. Apply the discriminant test to that primitive polynomial.
Formula notes
Polynomial content: g = GCF(a, b, c), so ax^2 + bx + c = g(Ax^2 + Bx + C)
- Find the coefficient GCF.
- Normalize the leading sign.
- Divide all coefficients equally.
- Keep the content outside.
Use the discriminant to construct or reject rational factors
For primitive Ax^2 + Bx + C, calculate D = B^2 - 4AC. A nonnegative perfect-square D gives rational roots. Reduce each root, convert it to a linear factor, and restore any leading scalar.
For 6x^2 + 5x + 1, D = 1 and roots -1/2 and -1/3 produce (2x + 1)(3x + 1). A non-square positive D gives irrational roots; a negative D gives complex roots.
Formula notes
Quadratic discriminant: D = B^2 - 4ACAn integer-coefficient quadratic factors into rational binomials exactly when D is a nonnegative perfect square
- D = 0: repeated root.
- Positive square: rational roots.
- Positive non-square: irrational roots.
- Negative: complex roots.
Verify factors by multiplication and expansion
Verify an integer by multiplying its prime powers and restoring the sign. Verify a quadratic coefficient by coefficient: (px + q)(rx + s) expands to prx^2 + (ps + qr)x + qs, including any outside scalar.
Reversed factor order and redistributed signs can still be equivalent. For an irreducible result, verify the extracted content and discriminant test, and state that the tested domain is the integers.
Formula notes
Expansion check: (px + q)(rx + s) = prx^2 + (ps + qr)x + qs
- Multiply back to n.
- Expand all factor terms.
- Compare a, b, and c.
- Accept equivalent order and signs.
- Name the tested domain.
Frequently asked questions
How do I find the prime factorization of an integer?
Choose Integer mode and enter the value. The calculator separates a negative sign as -1, finds the prime factors of the absolute value, groups repeated primes with exponents, and multiplies them back as a check.
Is 1 a prime number?
No. One is a multiplicative unit, not a prime or composite number. It has one positive divisor and no prime factors, which is why ordinary prime-factor rules begin with integers greater than 1.
Can zero be prime-factorized?
No. Every nonzero integer divides zero, so zero has infinitely many divisors and no unique finite prime factorization. The calculator reports that boundary explicitly.
How are the number of factors and factor pairs calculated?
If |n| = p1^e1 x ... x pk^ek, the positive divisor count is the product of (ei + 1). Positive factor-pair count is half that total, rounded up when |n| is a perfect square.
What does Euler's totient mean in the integer result?
Euler's totient phi(n) counts positive integers from 1 through n that are coprime to n. The calculator derives it exactly from the distinct prime factors of |n|.
How do I factor ax^2 + bx + c?
Choose Quadratic polynomial mode and enter signed integer coefficients. The calculator removes their content GCF, calculates the primitive discriminant, constructs exact binomials when its square root is an integer, and verifies the expansion.
References
These sources support the method or guidance used for Factoring Calculator. Verify time-sensitive rules at the source.
Try the calculator
Open Factoring Calculator, enter your scenario, and compare its supporting rows with this guide's method and checks.