Question

A polynomial \[ \varphi(s) = a_n s^n + a_{n-1} s^{n-1} + \cdots + a_1 s + a_0 \] of degree \(n>3\) with constant real coefficients has triple roots at \( s = -\sigma \). Which one of the following conditions must be satisfied?

• A. $\varphi(s)=0$ at all three values of $s$ satisfying $s^3+\sigma^3=0$
• B. $\varphi(s)=0,\quad \frac{d\varphi(s)}{ds}=0,\quad \frac{d^2\varphi(s)}{ds^2}=0$ at $s=-\sigma$
• C. $\varphi(s)=0,\quad \frac{d^2\varphi(s)}{ds^2}=0,\quad \frac{d^4\varphi(s)}{ds^4}=0$ at $s=-\sigma$
• D. $\varphi(s)=0,\quad \frac{d^3\varphi(s)}{ds^3}=0$ at $s=-\sigma$

Hint:

If a polynomial has a root of multiplicity $m$, then its first $m$ derivatives must be zero at that root.

Answer 1

The Correct Option is B

A triple root at \(s = -\sigma\) means the polynomial contains \((s+\sigma)^3\) as a factor. Step 1: Root multiplicity rule.
If the multiplicity of a root is 3, then: \[ \varphi(-\sigma)=0,\quad \varphi'(-\sigma)=0,\quad \varphi''(-\sigma)=0. \] Step 2: Why derivatives vanish.
For a root of multiplicity \(m\), the polynomial and its first \(m-1\) derivatives vanish at that point. Step 3: Compare with options.
Option (B) is exactly the required condition for a triple root. Other options are incorrect or impose wrong derivative requirements. Final Answer: Option (B)