Q:

Roots of the polynomial $P(z)=(\sum _{n=0}^5 a_nz^n)(\sum _{n=0}^9b_nz^n)$

Consider the polynomial $P(z)=(\sum _{n=0}^5 a_nz^n)(\sum _{n=0}^9b_nz^n)$ where $a_n,b_n\in \Bbb R$ $a_5\neq 0,b_9\neq 0$.
Then counting multiplicities we can conclude that $P(z)$ has :

at least  two real roots 
$14$ complex roots 
no real roots
$12$ complex roots.

Since an odd degree polynomial has at least one real root so $(\sum _{n=0}^5 a_nz^n)$ has one real root at least  and so does $(\sum _{n=0}^9 b_nz^n)$.
Hence $P(z)$ has at least two real roots counting multiplicities.So only $a$ is correct.
But my friend is arguing that $b$ is also true as each real number is a complex number.
But I think he is wrong as a complex number is one whose imaginary part is non-zero.
I am confused .Please help me to choose which one is correct.

A:

Your friend is correct.
Every real number is also complex.
The set of complex numbers is the set $\{a + bi | a,b \text{ are real}\}$.
There is no requirement that $b$ must be nonzero.
In a discussion, if someone says "the roots are complex", they probably mean "complex but non-real", so the context may allow you mentally insert the intended (but omitted) "non-real".
But in a formal question (e.g., a textbook exercise, an exam, a math competition) if "complex but non-real" is intended, it needs to be explicitly stated.

