II P(x)=(x-a)^2 Q(x), where Q(x) is a polynomial and Q(a) ≠0, we call x=a a double zero of the p

step 1 Hello there, in the following exercise we have this polynomial P that as you can observe the str

II P(x)=(x-a)^2 Q(x), where Q(x) is a polynomial and Q(a)...

II $P(x)=(x-a)^{2} Q(x),$ where $Q(x)$ is a polynomial and $Q(a) \neq 0,$ we call $x=a$ a double zero of the polynomial $P(x)$ (a) If $x=a$ is a double zero of a polynomial $P(x)$ show that $P(a)=P^{\prime}(a)=0$ (b) If $P(x)$ is a polynomial and $P(a)=P^{\prime}(a)=0$ show that $x=a$ is a double zero of $P(x)$

II $P(x)=(x-a)^{2} Q(x),$ where $Q(x)$ is a polynomial and $Q(a) \neq 0,$ we call $x=a$ a double zero of the polynomial $P(x)$
(a) If $x=a$ is a double zero of a polynomial $P(x)$ show that $P(a)=P^{\prime}(a)=0$
(b) If $P(x)$ is a polynomial and $P(a)=P^{\prime}(a)=0$ show that $x=a$ is a double zero of $P(x)$

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Key Concepts

Relationship between Zeros and Derivatives

When a polynomial has a zero of multiplicity m greater than one, not only does the polynomial vanish at that zero, but so do its first (m - 1) derivatives. In the case of a double zero (multiplicity 2), both the value of the polynomial and its first derivative at that point are zero. This relationship arises from the factorization and the application of the product rule during differentiation, and it provides a diagnostic tool for identifying multiple roots.

Factor Theorem

The Factor Theorem states that if a polynomial P(x) has a zero at x = a, then (x - a) is a factor of P(x). This theorem extends naturally to multiple roots, where if x = a is a zero of multiplicity greater than one, then higher powers of (x - a) will divide the polynomial without remainder. This is crucial for both verifying the existence of a zero and understanding its multiplicative structure.

Zero Multiplicity

Multiplicity of a zero refers to the number of times a particular root appears in the factorization of a polynomial. If a polynomial can be factored as (x - a)? times another polynomial that does not vanish at x = a, then a is a zero of multiplicity n. This concept is central to understanding how zeros behave, particularly how many times they 'repeat' and affect the shape of the polynomial's graph.

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