Answer to Question #17224 in Calculus for Ciaran Duffy

Question

Prove that if a is a double zero of the polynomial p(x) then p has a critical point
at x = a (that is p′(a) = 0).

Answer 1

Since a is a double zero of p(x), it can be factorized in the following
way:
p(x) = (x - a)^2 * q(x)
where q(x) is a polynomial. Then
p'(x) =
2(x - a) * q(x) + (x - a)^2 * q'(x) = (x - a) * (2*q(x) + (x - a) *
q(x)).
Here we see that the equation
p'(x) = 0
has a root x = a which
means that x = a is a critical point of p(x).