Question

Let \(\mathbb{R}\) denote the set of all real numbers. Consider the polynomial function \[ f : \mathbb{R} \to \mathbb{R} \] defined by \[ f(x) = \frac{d^{10}}{dx^{10}}\left((x^2 - 1)^{10}\right), \qquad \text{for all } x \in \mathbb{R}. \] Here, \[ \frac{d^{10}}{dx^{10}}\left((x^2 - 1)^{10}\right) \] is the \(10^{\text{th}}\) order derivative of the function \((x^2 - 1)^{10}\). Then which of the following statements is (are) TRUE?

• A. The coefficient of $x^8$ in the polynomial $f(x)$ is $(-10) \left( \frac{18!}{8!} \right)$
• B. The value of $f(1) + f(-1)$ is equal to $10! 2^{11}$
• C. The degree of the polynomial $f(x)$ is 10
• D. The constant term of the polynomial $f(x)$ is $- \left( \frac{10!}{5!} \right)$

Hint:

Recognizing the Rodrigues' formula structure ($n^{th}$ derivative of $(x^2-1)^n$) immediately simplifies problems involving specific values like $f(1)$ and $f(-1)$ due to the properties of Legendre polynomials.

Answer 1

The Correct Option is A