Greatest Common Divisor of a Polynomial and its First Derivative

Lemma 1:

Let a be a root of a polynomial P.

Then:

a is a multiple root of P if and only if a is also a root of P' (the first derivative of P)

Proof:

(1) Since a is a root of P, there exists a polynomial Q such that (see Theorem, here):

P = (x - a)Q

(2) Using the Product Rule of Derivatives (see Lemma 4, here), we know that:

P' = Q + (x-a)Q'

(3) But then (x-a) only divides P' if and only if it also divides Q.

QED

Corollary 1.1:

If a polynomial has only simple roots

Then:

Its first derivative does not share any of those roots.

Proof

(1) Assume that a polynomial P has only simple roots.

(2) Assume that a root a divides both P and P'

(3) Then by Lemma 1 above, a is a multiple root of P.

(4) But by step #1 this is impossible so we reject our assumption in step #2.

QED

Lemma 2:

If a polynomial P has only simple roots

Then P,P' are relatively prime.

Proof:

(1) Assume that P,P' are not relatively prime.

(2) Then, they have a common irreducible factor of degree 1.

[Since by definition, two polynomials are relatively prime if their only common factor is of degree 0]

(3) Then there exists a polynomial of the form X-a that divides both P and P' (See Thereom, here)

(4) Then from Lemma 1 above, a is a multiple root of P

(5) But this is impossible, so we reject our assumption in step #1.

QED

References

• Jean-Pierre Tignol, Galois' Theory of Algebraic Equations, World Scientific, 2001