fctr, sqfr

fctr(poly)

:: Factorize polynomial poly over the rationals.

sqfr(poly)

:: Gets a square-free factorization of polynomial poly.

return

list

poly

polynomial with rational coefficients

• Factorizes polynomial poly over the rationals. fctr() for irreducible factorization; sqfr() for square-free factorization.
• The result is represented by a list, whose elements are a pair represented as [[num,1],[factor,multiplicity],...].
• Products of all factor^multiplicity and num is equal to poly.
• The number num is determined so that (poly/num) is an integral polynomial and its content (GCD of all coefficients) is 1. (See section ptozp.)

[0] fctr(x^10-1);
[[1,1],[x-1,1],[x+1,1],[x^4+x^3+x^2+x+1,1],[x^4-x^3+x^2-x+1,1]]
[1] fctr(x^3+y^3+(z/3)^3-x*y*z);
[[1/27,1],[9*x^2+(-9*y-3*z)*x+9*y^2-3*z*y+z^2,1],[3*x+3*y+z,1]]
[2] A=(a+b+c+d)^2;
a^2+(2*b+2*c+2*d)*a+b^2+(2*c+2*d)*b+c^2+2*d*c+d^2
[3] fctr(A);
[[1,1],[a+b+c+d,2]]
[4] A=(x+1)*(x^2-y^2)^2; 
x^5+x^4-2*y^2*x^3-2*y^2*x^2+y^4*x+y^4
[5] sqfr(A);
[[1,1],[x+1,1],[-x^2+y^2,2]]
[6] fctr(A);
[[1,1],[x+1,1],[-x-y,2],[x-y,2]]

References

section ufctrhint.

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