Difference between revisions of "ExtendedGCD Command"
From GeoGebra Manual
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:Results are calculated by applying the [[w:Extended_Euclidean_algorithm|Extended Euclidean algorithm]]. | :Results are calculated by applying the [[w:Extended_Euclidean_algorithm|Extended Euclidean algorithm]]. | ||
| − | {{example| 1=<code><nowiki>ExtendedGCD(240,46)</nowiki></code> yields {<math>-9,47,2</math>}. (Plugging the result into the Bézout's identity we have: <math>-9 \cdot 240+47 \cdot 46=2</math>). }} | + | {{example| 1=<code><nowiki>ExtendedGCD(240,46)</nowiki></code> yields {<math>-9,47,2</math>}. <br> (Plugging the result into the Bézout's identity we have: <math>-9 \cdot 240+47 \cdot 46=2</math>). }} |
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:Results are calculated by applying the [[w:Extended_Euclidean_algorithm|Extended Euclidean algorithm]]. | :Results are calculated by applying the [[w:Extended_Euclidean_algorithm|Extended Euclidean algorithm]]. | ||
| − | {{example| 1=<code><nowiki>ExtendedGCD(x^2-1,x+4)</nowiki></code> yields {<math>1,-x+4,15</math>}. (Plugging the result into the Bézout's identity for polynomials we have: <math>1 \cdot (x^2-1) + (-x+4) \cdot (x+4) = 15</math>). }} | + | {{example| 1=<code><nowiki>ExtendedGCD(x^2-1,x+4)</nowiki></code> yields {<math>1,-x+4,15</math>}. <br> (Plugging the result into the Bézout's identity for polynomials we have: <math>1 \cdot (x^2-1) + (-x+4) \cdot (x+4) = 15</math>). }} |
{{notes|1=<div> | {{notes|1=<div> | ||
*The GCD of two polynomials is not unique (it's unique up to a scalar multiple). | *The GCD of two polynomials is not unique (it's unique up to a scalar multiple). | ||
*See also [[GCD Command]].</div>}} | *See also [[GCD Command]].</div>}} | ||
Latest revision as of 15:31, 2 May 2023
CAS Syntax
- ExtendedGCD( <Integer>,<Integer> )
- Returns a list containing the integer coefficients s, t of Bézout's identity as+bt= GCD(a,b) and the greatest common divisor of the given integers a and b.
- Results are calculated by applying the Extended Euclidean algorithm.
Example:
(Plugging the result into the Bézout's identity we have: -9 \cdot 240+47 \cdot 46=2).
ExtendedGCD(240,46) yields {-9,47,2}. (Plugging the result into the Bézout's identity we have: -9 \cdot 240+47 \cdot 46=2).
- ExtendedGCD( <Polynomial>, <Polynomial> )
- Returns a list containing the polynomial coefficients S(x), T(x) of Bézout's identity for polynomials A(x)S(x) + B(x)T(x) = GCD(A(x), B(x)) and the greatest common divisor of the given polynomialsA(x) and B(x).
- Results are calculated by applying the Extended Euclidean algorithm.
Example:
(Plugging the result into the Bézout's identity for polynomials we have: 1 \cdot (x^2-1) + (-x+4) \cdot (x+4) = 15).
ExtendedGCD(x^2-1,x+4) yields {1,-x+4,15}. (Plugging the result into the Bézout's identity for polynomials we have: 1 \cdot (x^2-1) + (-x+4) \cdot (x+4) = 15).
Notes:
- The GCD of two polynomials is not unique (it's unique up to a scalar multiple).
- See also GCD Command.
