Let n be a positive integer. We say that a polynomial P(x) has a zero of multiplicity n at a point a if P(x) = (x – a)"Q(x), for some other polynomial Q(x) with the property that Q(a) 0. Prove, by induction on n, that if P has a zero of multiplicity n at а, then P(a) = P'(a) : p(n-1)(a) = 0 and p(w) (a) # 0. (Hint: Show that if P(x) has a zero of multiplicity k + 1, then P' has a zero of multiplicity k.)

Let n be a positive integer. We say that a polynomial P(x) has a zero of multiplicity n at a point a if P(x) = (x – a)"Q(x), for some other polynomial Q(x) with the property that Q(a) 0. Prove, by induction on n, that if P has a zero of multiplicity n at а, then P(a) = P'(a) : p(n-1)(a) = 0 and p(w) (a) # 0. (Hint: Show that if P(x) has a zero of multiplicity k + 1, then P' has a zero of multiplicity k.)

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN:9781305071742

Author:James Stewart, Lothar Redlin, Saleem Watson

Publisher:James Stewart, Lothar Redlin, Saleem Watson

Chapter3: Polynomial And Rational Functions

Section3.CT: Chapter Test

Problem 9CT

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