Polynomial of Degree 98: Value of $p(100)$

[juantheron]

If $p(x)$ be a polynomial of degree $98$ such that $\displaystyle p(x) = \frac{1}{x}$ for $x=1,2,3,...,98$

Then value of $p(100)=$

 

[CaptainBlack]

If $p(x)$ be a polynomial of degree $98$ such that $\displaystyle p(x) = \frac{1}{x}$ for $x=1,2,3,...,98$

Then value of $p(100)=$

Is there no other condition?

The polynomial \(p(x)\) being of degree 98 has 99 degrees of freedom, there are 98 constraints which leaves a degree of freedom. Now if we knew that it was a monic polynomial we could answer:

\(p(100)=99!\)

(that is assuming I have the algebra right, the method is related to Opalg's approach)

CB

 

[Opalg]

I wonder whether this problem should be:

If $p(x)$ be a polynomial of degree $98$ such that $\displaystyle p(x) = \frac{1}{x}$ for $x=1,2,3,...,\color{red}{99}$

Then value of $p(100)=$

In that case, let $q(x) = xp(x)-1$. Then $q(x)$ is a polynomial of degree 99, and $q(x)=0$ for $x=1,2,3,\ldots,99$. By the factor theorem, $q(x) = k(x-1)(x-2)\cdots(x-99)$ for some constant $k$. Also, $q(0)=-1$, and therefore $k=1/99!$. It follows that $q(100) = 1$, from which $p(100) = 1/50$.

 

[juantheron]

Sorry caption and opalg i have missed that

 

[CellCoree]

I would first clarify that the question is asking for the value of $p(100)$ assuming that the polynomial $p(x)$ is defined as $\displaystyle p(x) = \frac{1}{x}$ for all integers from $1$ to $98$.

Based on this assumption, the value of $p(100)$ would be undefined because the polynomial $p(x)$ is not defined for $x=100$. This is because the polynomial has a degree of 98, meaning it does not have any terms with $x^{99}$ or higher. Therefore, it cannot be used to find the value of $p(100)$.

If the question is asking for the value of $p(100)$ assuming that the polynomial $p(x)$ is defined as $\displaystyle p(x) = \frac{1}{x}$ for all real numbers from $1$ to $98$, then the value of $p(100)$ would still be undefined because the polynomial would still not have a term with $x^{99}$ or higher, making it unable to be used for $x=100$.

In summary, the value of $p(100)$ cannot be determined based on the given information and assumptions. It is important to ensure that the polynomial is properly defined and that the degree of the polynomial is considered when attempting to find the value of $p(100)$.