In control theory, where the transfer function of a linear system is

$$H(s)=\frac{c_k s^k+\dots c_1s+c_0}{a_n s^n+\dots a_1s+a_0}$$

if $k\le n$ and the denominator has roots whose real parts is negative, then the system is called stable.

So, let's call a polynomial stable if the real parts of all its roots are negative.

Assuming that $p$ and $q$ are two stable polynomials with positive $a_n$ and $b_m$ as follows

$$p(x)=a_nx^n+\dots+a_1x+a_0$$ $$q(x)=b_mx^m+\dots+b_1x+b_0$$

Then, can one claim that $p(x)+q(x)$ is necessarily stable too?


No, consider for example $p(x)=(x+1)^3$ and $q(x)=x+11$, then $p+q$ has roots $\pm 2i$ with real part $0\,$. Using $q(x)=x+a$ with $a \gt 11\,$, instead, gives a $p+q$ with two complex roots having strictly positive real part.


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