# Answer to Question #17224 in Calculus for Ciaran Duffy

Question #17224

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).

at x = a (that is p′(a) = 0).

Expert's answer

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).

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).

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