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gcdex

extended Euclidean algorithm for polynomials

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

gcdex(A, B, x, 's', 't')

gcdex(A, B, C, x, 's', 't')

Parameters

A, B, C

-

polynomials in the variable x

x

-

variable name

s, t

-

(optional) unevaluated names

Description

• 

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 degrees<degreeBg and degreet<degreeAg. 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 degrees<degreeBg. If degreeCg<degreeAg+degreeBg then the polynomial t will satisfy degreet<degreeAg. 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.

Examples

gcdexx31&comma;x21&comma;x&comma;&apos;s&apos;&comma;&apos;t&apos;

x1

(1)

s&comma;t

1,x

(2)

gcdexx2&plus;a&comma;x21&comma;x2a&comma;x&comma;&apos;s&apos;&comma;&apos;t&apos;

s&comma;t

1+aa+1,2aa+1

(3)

gcdex1&comma;x&comma;12x&plus;4x2&comma;x&comma;&apos;s&apos;&comma;&apos;t&apos;

s&comma;t

1,4x2

(4)

gcdexx21&comma;x31&comma;x&comma;x&comma;&apos;s&apos;&comma;&apos;t&apos;

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

gcdx21&comma;x31

x1

(5)

Compatibility

• 

The gcdex command was updated in Maple 2018.

See Also

degree

gcd

Gcdex

igcdex