If I told you a certain polynomial P(x) had a double root (only one!), how could you go about finding it, WITHOUT trying to find every root? Also, the EXACT value of the root is sought; not an approximation.

NB. Roots may be any kind --real or complex-- but they are all different, with multiplicity "1", except for one that has multiplicity "2".

(In reply to A most general solution by e.g.)

Yours is the right track, although you probably mean GCD instead of GCM (whatever that is).   The solution comes from the Greatest Common Divisor (GCD), also known as the Highest Common Factor (HCF), of P(x) and P'(x), which will be a polynomial of degree 1.  The degree of the GCD will be 1 because P'(x) can have no other root in common with P(x) besides the one root of multiplicity 2. Hence the root sought will be the easily-found root of a linear equation.  The method then is to apply Euclid's Algorithm for polynomials to find the GCD of P(x) and P'(x) and then to find the unique root of the GCD. This is the path my previous comment should have led me to had I been thinking better.

Posted by Richard on 2006-05-08 16:38:38