# Polynomial factorization

The calculator finds all factors of a polynomial with rational coefficients

The calculator below finds all irreducible factors of a polynomial with rational coefficients. To better understand how it works, switch on the 'Show details' toggle and read the calculator's description. #### Polynomial factorization with rational coefficients

Input polynomial

Solution

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## Rational polynomial factorization procedure1

• Convert input polynomial in Q[x] to primitive polynomial in Z[x]
• Find all square factors using Yun square-free factorization algorithm
• For each square-free factor of degree greater than 1, do the following steps
• If leading coefficient is not equal to 1 then transform it to monic one using formula:
, where
v(y) - transformed monic polynomial,
u(x) - original polynomial,
an - leading coefficient of u(x),
x = any
• Find irreducible factors of v(y)=v1v2...vr in finite field Fp[x]
• Find minimal prime number which is not divisor of v(y) discriminant
• Use Hensel lifting to raise the finite field order of the factorization to the upper limit
• Determine upper limit of target factors coefficients by formula:
, where
- maximum absolute value of polynomial coefficients (polynomial height)
• Perform hensel lifting times
• Check the factors by division v(y)/vi in Z[x], remove invalid factors
• Invert monic polynomial transformation using formula:

pp - primitive part function, which removes a content form an input polynomial

1. Joel S. Cohen, Computer Algebra and Symbolyc Computation: Mathematical Methods, par. 9. Polynomial Factorization

2. Donald Knuth, The Art of Computer Programming vol.2, par. 4.6.2 Factorization of Polynomials

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PLANETCALC, Polynomial factorization