Show that, if a polynomial p(x) with real coefficients has m distinct real roots,then p´(x) has m-1 distinct real roots
Thanks for your help
Let a(1), a(2), ..., a(m) be the distinct real roots of p(x).
Now, apply Rolle's Theorem to p(x) on [a(k), a(k+1)] for k = 0, 1, ..., m-1:
==> p'(c) = 0 for some c in (a(k), a(k+1)) for k = 0, 1, ..., m-1:
==> p'(x) has m-1 distinct real roots.
I hope this helps!
Copyright © 2024 1QUIZZ.COM - All rights reserved.
Answers & Comments
Verified answer
Let a(1), a(2), ..., a(m) be the distinct real roots of p(x).
Now, apply Rolle's Theorem to p(x) on [a(k), a(k+1)] for k = 0, 1, ..., m-1:
==> p'(c) = 0 for some c in (a(k), a(k+1)) for k = 0, 1, ..., m-1:
==> p'(x) has m-1 distinct real roots.
I hope this helps!