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gcdex -- the extended Euclidean algorithm for polynomials

Introduction

gcdex(p, q, x) regards p and q as univariate polynomials in x and returns their greatest common divisor as a linear combination of p and q.

Call(s)

gcdex(p, q <, x>)
gcdex(f, g, x)

Parameters

p, q - polynomials of type DOM_POLY
f, g - polynomial expressions
x - an indeterminate: an identifier or an indexed identifier

Returns

a sequence of three polynomials, or a sequence of three polynomial expressions, or FAIL.

Overloadable:

p, q

Related Functions

factor, div, divide, gcd, ifactor, igcd, igcdex, ilcm, lcm, mod, poly

Details

Example 1

The greatest common divisor of two univariate polynomials in extended form can be computed as follows:

>> gcdex(poly(x^3 + 1), poly(x^2 + 2*x + 1))
        poly(x + 1, [x]), poly(1/3, [x]), poly(- 1/3 x + 2/3, [x])

For multivariate polynomials, an indeterminate must be specified:

>> gcdex(poly(x^2*y), poly(x + y), x)
                          / 1       \      / /   1  \     1      \
        poly(1, [x]), poly| --, [x] |, poly| | - -- | x + -, [x] |
                          |  3      |      | |    2 |     y      |
                          \ y       /      \ \   y  /            /
>> gcdex(poly(x^2*y), poly(x + y), y)
                                /   1       \      / 1      \
              poly(1, [y]), poly| - --, [y] |, poly| -, [y] |
                                |    3      |      \ x      /
                                \   x       /
>> gcdex(x^3 + a, x^2 + 1, x)
                                          2
                             a + x   1 - x  - a x
                          1, ------, ------------
                              2          2
                             a  + 1     a  + 1

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