If P(x)=(x-a)^2 Q(x), where Q(x) is a polynomial, we call...

If $P(x)=(x-a)^{2} Q(x),$ where $Q(x)$ is a polynomial, 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 $P(x)$

If $P(x)=(x-a)^{2} Q(x),$ where $Q(x)$ is a polynomial, 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

Multiplicity of Roots

Multiplicity of a root refers to the number of times a particular solution appears as a factor in a polynomial. If x = a is a double zero (or double root), that means it is repeated twice as a factor. In general, a root of multiplicity m implies that the polynomial can be written as (x - a)^m multiplied by some other polynomial where (x - a) does not divide that polynomial.

Factor Theorem

The Factor Theorem states that if a polynomial P(x) has a root at x = a, then (x - a) is a factor of P(x). Conversely, if (x - a) is a factor, then P(a) equals zero. This theorem is the foundation for connecting the value of the polynomial at a specific point with its factorization.

Relationship between Derivatives and Multiple Zeros

When a root has multiplicity greater than one, not only does the polynomial evaluate to zero at that point, but its derivative does as well. For example, if x = a is a double zero (i.e., a root of multiplicity two), differentiating the polynomial will still include (x - a) as a factor, making the derivative also vanish at x = a. This property is key in distinguishing single roots from multiple roots.

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