Analyzing Polynomial Roots: $(s+1)(s+2)(s+3)(s+4)$

The question asks us to identify the correct statement about the roots of the polynomial $P(s) = (s+1)(s+2)(s+3)(s+4)$. The roots of a polynomial are the values of $s$ that make the polynomial equal to zero, i.e., $P(s) = 0$. We need to determine how many of these roots have positive real parts.

Finding the Roots

The polynomial is given in a factored form, which makes finding the roots straightforward. We set the polynomial equal to zero:

$ (s+1)(s+2)(s+3)(s+4) = 0 $

For the product of factors to be zero, at least one of the factors must be zero. We find the roots by setting each factor to zero:

• $s+1 = 0 \implies s = -1$
• $s+2 = 0 \implies s = -2$
• $s+3 = 0 \implies s = -3$
• $s+4 = 0 \implies s = -4$

So, the roots of the polynomial are $s = -1, -2, -3,$ and $-4$.

Analyzing the Real Parts of the Roots

Now, we examine the real parts of these roots:

• The root $s = -1$ has a real part of $-1$.
• The root $s = -2$ has a real part of $-2$.
• The root $s = -3$ has a real part of $-3$.
• The root $s = -4$ has a real part of $-4$.

All the roots are real numbers, and all of them have negative real parts.

Conclusion on Positive Real Parts

Since all the real parts of the roots $(-1, -2, -3, -4)$ are negative, there are no roots with positive real parts.

Therefore, the correct statement is that there are no roots with positive real parts.