Let f(x) ∈𝐑[x]. Suppose that f(a)=0 but f^'(a) ≠0, where f^'(x) is the derivative of f(x). Show

Let f(x) ∈𝐑[x]. Suppose that f(a)=0 but f^'(a) ≠0, where...

Key Concepts

Multiplicity of a Zero

The concept of the multiplicity of a zero refers to the number of times a particular root appears in the factorization of a polynomial. When a polynomial f(x) is expressed as (x - a)^m * g(x), where g(a) ? 0, the number m is called the multiplicity of the zero a. A zero of multiplicity 1 is called a simple zero, indicating that it appears only once in the factorization.

Relationship between Derivatives and Multiplicity

There is a direct connection between the multiplicity of a zero and the behavior of the polynomial's derivative at that point. If a is a zero of multiplicity m, then the first m - 1 derivatives of the polynomial evaluated at a are zero, while the mth derivative is nonzero. In the case of a simple zero (multiplicity 1), the function f(x) satisfies f(a) = 0 and its derivative f'(a) ? 0. This relationship is a useful tool for identifying the simplicity of a root.

Polynomial Factorization and Local Behavior

Factorization of the polynomial into (x - a)^m * g(x), with g(a) ? 0, allows us to analyze its behavior near the zero a. This local factorization clarifies that the nonzero derivative f'(a) is a consequence of the nonvanishing factor g(a), confirming that the multiplicity m must equal 1. This approach links the algebraic structure of the polynomial with its analytic properties.

Transcript

0:00 Hello there.

00:02 For the photowin exercise, we know that the polynomial with a zero of multiplicity m at the point x equals to a have this structure.

00:11 So this represents these kind of polynomials which have multiplicity m and the root in x equals to a.

00:20 It is clear that if we evaluate this function at a, we have that this equals to zero and it has multiplicity m because this root repeat m times.

00:30 And so this part here represent that multiplicity of the root or of the zero.

00:39 And this hx is an other polynomial that is non -zero when we evaluated at a.

00:46 Okay, so having this general structure of a polynomial with a zero at a and of multiplicity m, we need to explain why the p derivative, this notation here represents a p derivative of this.

01:00 Function it's equal to zero for all the values of p from 1 to m minus 1 okay so let's first observe what happened with the first derivative so the first derivative of this function here is evaluated at a the first derivative evaluated at a of this function is m times x minus a m minus 1 h x and plus x minus a m times h prime a x and this is evaluated at x equals to a and when we evaluate we have that this is equals to zero and this also equals to zero so this derivative is equals to zero then if we take another derivative for this function and we evaluate at a now we obtain the following expression so now we obtain here m then m minus a m minus one sorry times a a minus one here we have x minus a m minus two times h x plus plus two times two times we can put it this directly yes two times m h prime x and x minus a m minus 1 and the last term is just plus h double prime x and here x minus a m okay so again here we you can observe that if we evaluate at a x equals to a then this term is going to be zero this is also zero and this is also zero but at this point and want you to notice something else and is that this derivative the second derivative can be written as a polynomial in terms of x minus a okay so look that f double prime of a evaluated at a it's equal to a polynomial of this kind so we have alpha 1 x minus a m minus 2 plus alpha 2 x minus a m minus 1 and here plus alpha 3 x minus a m okay so this is like the general structure for a second derivative right we don't care what are this constant we know that it is expressed that in some terms of h and h prime and h double prime and some other coefficients that result from the multiplication of these terms so for example alpha 1 is m times n minus 1 and so on so we can write this as a polynomial.

04:48 This constant here, it doesn't matter so far we're interested in these monomials that appear...

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