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For the first calling sequence (when the number of parameters is less than six), gcdex applies the extended Euclidean algorithm to compute unique polynomials s, t, and g in x such that $sA\&plus;tB\&equals;g$ where g is the monic GCD (Greatest Common Divisor) of A and B. The results computed satisfy $\mathrm{degree}\left(s\right)<\mathrm{degree}\left(\frac{B}{g}\right)$ and $\mathrm{degree}\left(t\right)<\mathrm{degree}\left(\frac{A}{g}\right)$. The GCD g is returned as the function value.

If arguments s and t are specified, they are assigned the cofactors.

In the second calling sequence, gcdex solves the polynomial Diophantine equation $sA\&plus;tB\&equals;C$ for polynomials s and t in x. Let g be the GCD of A and B. The input polynomial C must be divisible by g; otherwise, an error message is displayed. The polynomial s computed satisfies $\mathrm{degree}\left(s\right)<\mathrm{degree}\left(\frac{B}{g}\right)$. If $\mathrm{degree}\left(\frac{C}{g}\right)<\mathrm{degree}\left(\frac{A}{g}\right)+\mathrm{degree}\left(\frac{B}{g}\right)$ then the polynomial t will satisfy $\mathrm{degree}\left(t\right)<\mathrm{degree}\left(\frac{A}{g}\right)$. The NULL value is returned as the function value.

In this case, s and t are not optional.

Note that if the input polynomials are multivariate then, in general, s and t will be rational functions in variables other than x.

$\mathrm{gcdex}\left(x^3-1\&comma;x^2-1\&comma;x\&comma;'s'\&comma;'t'\right)$

$x-1$

$s,t$

$1,-x$

$\mathrm{gcdex}\left(x^2+a\&comma;x^2-1\&comma;x^2-a\&comma;x\&comma;'s'\&comma;'t'\right)$

$s,t$

$-\frac{-1+a}{a+1},\frac{2a}{a+1}$

$\mathrm{gcdex}\left(1\&comma;x\&comma;1-2x+4x^2\&comma;x\&comma;'s'\&comma;'t'\right)$

$s,t$

$1,4x-2$

$\mathrm{gcdex}\left(x^2-1\&comma;x^3-1\&comma;x\&comma;x\&comma;'s'\&comma;'t'\right)$

Error, (in `gcdex/diophant`) the Diophantine equation has no solution

$\gcd \left(x^2-1\&comma;x^3-1\right)$

$x-1$

The gcdex command was updated in Maple 2018.