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Applications of the Mean Value Theorem: Roots of Polynomials

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Show that if the roots of the polynomial p are all real, then the roots of p' are all real. If, in addition, the roots of p are all simple, then the roots of p' are all simple.

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Real and Simple Roots of Polynomials are investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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The solution:

Let the roots of the polynomial equation p(x)=0 are real. Let a and b are any two roots of p(x). So, We have the following conditions satisfied by p(x)

1. p(x) is continuous as it is a polynomial
2. p(x) is differentiable for the same reason
3. p(a)=p(b)

Hence ...

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