MagicMath: Polynomial GCD

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Polynomial Greatest Common Divisor

The greatest common divisor (gcd), of two polynomials is the polynomial of highest degree that divides both polynomials. If the polynomial D(x) is the gcd of the numbers P(x) and Q(x), we write

gcd ( P(x), Q(x) ) = D(x)

For example,

gcd ( (x^4 - 1), (x^3 + 1) ) = x + 1

The leading coefficient of the gcd is kept positive.

When simplifying a rational function, the gcd of the numerator and the denominator polynomials is needed. Then the gcd can be removed from the numerator and denominator.

On this page, you can compute gcd of polynomails with integer coefficients. p>

Enter P(x) and Q(x) to see the gcd:

The gcd of more than two polynomials is defined naturally as the greatest polynomial that devides all the given polynomials. You can compute the gcd of many polynomials by using gcd of two polynomials repeatedly.

For example

gcd ( H(x), P(x), Q(x) )

can be done by computing

gcd ( H(x), P(x) ) = D(x)

then computing

gcd ( D(x), Q(x) )