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".
Assume that P(x) has a double root A.
Let P(x)= (x-A)²Q(x), where Q(A)≠0,
By Euclid 's algorithm we know that: P(x) will have a double root, when:
gcd{P(x), P'(x)} = x-A,
We now observe that: .
P'(x)= (x-A)[2Q(x)+(x-A)Q'(x)].
So that: gcd{P(x), P'(x)} = x-A
Consequently, we can now state with certainty that P(x) has a double root.
Edited on December 31, 2022, 1:54 am
Puzzle Solution
