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The Jacal command factor takes as input an integer and returns a
list of the prime numbers that divide it, each occurring with the
appropriate multiplicity in the list. If the number is negative, the
list will begin with -1.
The results of the factor command are shown in a special
factored format, which appears as the product of the factors.
e0 : factor(120);
3
e0: 2 3 5
e1 : factor(-120);
3
e1: -1 2 3 5
Given a univariate ratio of polynomials polyratio, returns a matrix of factors and exponents.
As above, the results are shown in factored form.
e2 : factor((14*x^4-10/68*x^-5)/(5*x^2+1));
9
-5 + 476 x
e2: ------------------
2 5
2 17 (1 + 5 x ) x
e3 : (14*x^4-10/68*x^-5)/(5*x^2+1);
9
-5 + 476 x
e3: --------------
5 7
34 x + 170 x
e4 : (476*x^9-5)/(34*(5*x^2+1)*x^5);
9
-5 + 476 x
e4: --------------
5 7
34 x + 170 x
e5 : factor(x*y);
e5: y x
e6 : factor((x+a)*(y^4-z));
4
e6: -1 (a + x) (- y + z)
e7 : factor((x+u*a^3)*(y^4-z));
3 4
e7: -1 (a u + x) (- y + z)
e8 : factor((x+u*a^3)^2*(y^4-z)/((x+1)*(u^2-v^2)));
4 3 2
(- y + z) (a u + x)
e8: -------------------------
(1 + x) (- u + v) (u + v)
e9 : factor(200*(-1*x+1+y)*(u-r^6)*(21*x+2-t^4));
3 2 6 4
e9: 2 5 (- r + u) (1 - x + y) (2 - t + 21 x)
e10 : factor(2*(a+u)*(-v+b)*(a*x+y)^2);
2
e10: -1 2 (a + u) (- b + v) (a x + y)
e11 : factor(2*(a+u)*(-v+b)*(a*x+y)^2/((u^2-v^2)*(11*x+55)));
2
2 (a + u) (- b + v) (a x + y)
e11: ------------------------------
11 (5 + x) (- u + v) (u + v)
e12 : factor(2*(a+u)*(-v+b)*(a*x+y)^2/((u^2-v^2)*x^4*(11*x+55)));
2
2 (a + u) (- b + v) (a x + y)
e12: -------------------------------
4
11 (5 + x) (- u + v) (u + v) x
e13 : factor((c^3*u+b*a)*(b*b*a+v*p^2*q^2*c));
3 2 2 2
e13: (a b + c u) (a b + c p q v)
e14 : factor((2*z+y-x)*(y^3-a*x^2)*(b*z^2+y));
2 2 3
e14: (- x + y + 2 z) (y + b z ) (- a x + y )
e15 : factor((a*a*b*z+d)*(2*a*b*b*z+c));
2 2
e15: (d + a b z) (c + 2 a b z)
e16 : factor((a*a*b*z+d)*(2*a*b*b*z+c)*((u+a)*x+1));
2 2
e16: (1 + (a + u) x) (d + a b z) (c + 2 a b z)
e17 : factor((c*z+a)*(a*z+b)*(b*z+c));
e17: (b + a z) (c + b z) (a + c z)
e18 : factor((a*a*b*(x+w)*z+d)*(2*a*b*b*z+c));
2 2 2
e18: (d + (a b w + a b x) z) (c + 2 a b z)
e19 : factor(((x+w)^2*z-u*d)*(-2*a*b*z+c));
2 2
e19: -1 (- c + 2 a b z) (- d u + (w + 2 w x + x ) z)
e20 : factor((-200*%i*x-c)*(x-d-z^5)/(a*(b^3-(a+u)*z)));
5
-1 (c + 200 %i x) (d - x + z )
e20: ------------------------------
3
a (- b + (a + u) z)
The rest of this section documents commands from the factoring package. To use this package, execute the following command from the JACAL prompt:
require("ff");
Several of these commands return a matrix. The first column contains the factors and the second column contains the corresponding exponent.
Given a primitive univariate polynomial poly, calculate the square free factorisation of poly. A primitive polynomial is one with no factors (other than units) common to all its coefficients.
Given a monic polynomial poly, a prime p, and a positive
integer m, calculate the square free factorisation of poly
in GF(p^m)[x]. If m is not supplied, 1 is assumed.
e0 : ffsff(x^5+x^3+1, 53);
[ 2 3 ]
[16 - 22 x + 26 x + x 1]
e0: [ ]
[ -13 + x 2]
Given a square-free univariate polynomial poly and an integer power of a prime, q, returns (as a bunch) the irreducible factors of poly.
e2 : berl(x^5+x^3+2, 53);
2 2
e2: [1 + x, 5 - 26 x + x , 11 + 25 x + x ]
Returns the partial fraction expansion of a rational univariate polynomial polyratio. The denominator of polyratio must be square free. This code is still being developed.
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