Polynomial fractions

Calculating the greatest common divisor (GCD) is one of the very useful operations to simplify the fractions of polynomials. Other common requirements are extracting the numerator, the denominator, and the highest power. Here is the function demonstrating all of these: $ maxima -q (%i1) gcd(x^3 + 3*x^2 + 3*x + 1, x^2 + 3*x + 2); (%o1) x+1 (%i2) string(ezgcd(x^3 + 3*x^2 + 3*x + 1, x^2 + 3*x + 2)); (%o2) [x+1,x^2+2*x+1,x+2] (%i3) string(denom((x + 1)^-3 * (1 - x)^2)); (%o3) (x+1)^3 (%i4) string(num((x + 1)^-3 * (1 - x)^2)); (%o4) (1-x)^2 (%i5) hipow(expand((x + 1)^3 + (1 - x)^3), x); (%o5) 2 (%i6) quit();

Note that the ezgcd() function lists out the remainder polynomials, along with the GCD.

Polynomial fractions can be differentiated using the powerful ratdiff(): $ maxima -q (%i1) string(ratdiff((x + 1)^-1 * (1 - x)^2, x)); (%o1) (x^2+2*x-3)/(x^2+2*x+1) (%i2) string(ratdiff(1 / (x + 1), x)); (%o2) -1/(x^2+2*x+1) (%i3) string(ratdiff((x^2 - 1) / (x + 1), x)); (%o3) 1 (%i4) quit();

And ratsubst() is a powerful expression substitution function, with intelligence. It can dig into the expression to simplify complicated expressions, including trigonometric ones. Check out the %i5, for one of its powerful applications. ratsubst(<new>, <old>, <expr>) replaces the <old> expression by the <new> expression in the complete expression <expr>: $ maxima -q (%i1) string(ratsubst(u, x^2, x^3 + 3*x^2 + 3*x + 1)); (%o1) (u+3)*x+3*u+1 (%i2) string(ratsubst(u, x^2, (x+1)^3)); (%o2) (u+3)*x+3*u+1 (%i3) string(ratsubst(u, x^2, (x+1)^4)); (%o3) (4*u+4)*x+u^2+6*u+1 (%i4) string(ratsubst(u, x - 1, x^4 - 2*x^2 + 1)); (%o4) u^4+4*u^3+4*u^2 (%i5) string(ratsubst(sin(x)^2, 1 - cos(x)^2, cos(x)^4 - 2*cos(x)^2 + 1)); (%o5) (%i5) quit(); -

sin(x)^4