Returns the degree of polynomial or equation poly in variable var.
Returns the total-degree, the degree of its highest degree monomial, of polynomial or equation poly.
(%0028) degree(a*x*x + b*y*x + c*y*y + d*x + e*y + f, y); %0028: 2 (%0029) degree(a*x*x + b*y*x + c*y*y + d*x + e*y + f); %0029: 3
Note that all roots and radicals count as degree 1.
The command coeff is used to determine the coefficient of a
certain power of a variable in a given polynomial. Here poly is a
polynomial and var is a variable. If the optional third argument
is omitted, then Jacal returns the coefficient of the variable var
in poly. Otherwise it returns the coefficient of var^deg in
poly. The function coeffs returns a list of all of the
coefficients. For example,
(%0009) coeff((x + 2)^4, x, 3);
%0009: 8
(%0010) (x + 2)^4;
2 3 4
%0010: 16 + 32 x + 24 x + 8 x + x
(%0011) coeff((x + 2)^4, x);
%0011: 32
(%0012) coeffs((x + 2)^4, x);
%0012: [16, 32, 24, 8, 1]
The function poly provides an inverse to the function
coeffs, allowing one to recover a polynomial from its vector or
list of coefficients.
(%0013) poly(y, [16, 32, 24, 8, 1]);
2 3 4
%0013: 16 + 32 y + 24 y + 8 y + y
(%0014) poly(y, 16, 32, 24, 8, 1);
2 3 4
%0014: 16 + 32 y + 24 y + 8 y + y
The function poly returns the expression equal to 0 in equation
eqn. Be aware that the sign and scaling of the returned
polynomial will not necessarily match those in the equation creating
eqn.
(%0015) 2*a = 4*c; %0015: 0 = - a + 2 c (%0016) poly(%0015); %0016: - a + 2 c
Returns a list of content and primitive part of a polynomial with respect to the variable. The content is the GCD of the coefficients of the polynomial in the variable. The primitive part is poly divided by the content.
content(2*x*y+4*x^2*y^2,y);
2
%0017: [2 x, y + 2 x y ]
The command divide treats divident and divisor as
polynomials in the variable var and returns a pair
‘[quotient, remainder]’ such that dividend
= divisor * quotient + remainder. If the third
argument var is omitted Jacal will choose a variable on its own
with respect to which it will do the division. In particular, of
dividend and divisor are both numerical, one can safely omit
the third argument.
(%0018) divide(x^2+y^2,x-7*y^2,x);
2 2 4
%0018: [x + 7 y , y + 49 y ]
(%0019) divide(-7,3);
%0019: [-2, -1]
(%0020) divide(x^2+y^2+z^2,x+y+z);
2 2
%0020: [- x - y + z, 2 x + 2 x y + 2 y ]
(%0021) divide(x^2+y^2+z^2,x+y+z,y);
2 2
%0021: [- x + y - z, 2 x + 2 x z + 2 z ]
(%0022) divide(x^2+y^2+z^2,x+y+z,z);
2 2
%0022: [- x - y + z, 2 x + 2 x y + 2 y ]
Returns poly1 reduced with respect to poly2 (or eqn) and var. If poly2 is univariate, the third argument is not needed.
Returns poly1 with all the coefficients taken modulo n.
Returns poly1 with all the coefficients taken modulo the current modulus.
If the modulus (n or the current modulus) is negative, then the results use symmetric representation.
(%0023) x^4+4 mod 3;
4
%0023: 1 + x
(%0024) x^4+4 mod x^2=2;
%0024: 8
(%0025) mod(x^3*a*7+x*8+34, -3);
3
%0025: 1 - x + a x
(%0026) mod(5,2);
%0026: 1
(%0027) mod(x^4+4,x^2=2,x);
%0027: 8
The Jacal function gcd takes as arguments two polynomials with
integer coefficients and returns a greatest common divisor of the two
polynomials. This includes the case where the polynomials are
integers.
(%0028) gcd(x^4-y^4,x^6+y^6);
2 2
%0028: x + y
(%0029) gcd(4,10);
%0029: 2
Here poly is a polynomial and var is a variable. This function returns the square of the product of the differences of the roots of the polynomial poly with respect to the variable var.
(%0030) discriminant(x^3 - 1, x); %0030: -27
The function resultant returns the resultant of the polynomials
poly_1 and poly_2 with respect to the variable
var.
(%0031) resultant(x^2 + a, x^3 + a, x);
2 3
%0031: a + a
Returns the list of equations formed by equating each coefficient of variable var^n in z1 to the corresponding coefficient of var^n in z2. z1 and z2 can be polynomials or ratios of polynomials.
Returns the polynomial decomposition of poly_1 with respect to
var. Note that decompose is not currently working.