# Doubts regarding meaning of certain mathematical statement

I am posting this question to clarify some doubts I have regarding meaning of certain mathematical statements. So in the question below the polynomial $P$ is a product of two odd degree polynomials. From this we can conclude that it has at lest $2$ real roots. Further it has $14$ complex roots (Fundamental theorem of algebra). My question is why is statement $(D)$ not correct?

Is $(D)$ equivalent to the statement "$P(z)$ has exactly 12 complex roots"

Consider the polynomial $$P(z)=\left(\sum\limits_{n=0}^5 a_nz^n\right)\left(\sum\limits_{n=0}^9 b_n z^n \right)$$ $a_nb_n \in\mathbb{R}, \ a_5,b_9 \neq 0$

Then counting multiplicity we can conclude that $P(z)$ has

$(A)$ at least two real roots

$(B)$ $14$ complex roots

$(C)$ no real roots

$(D)$ 12 complex roots

Answer provided: (A) $\&$ (B)

• your interpretation of D seems to be correct. – Anurag A Jan 16 '17 at 8:42
• If all coefficients of $P(z)$ are real, so we can say at the first sight that, all roots are conjugates. – Nosrati Jan 16 '17 at 8:48
• Presumably "$a_5,b_5\ne 0$" should be $a_5,b_9\ne 0$? Otherwise you can't be sure there are more than one real root and 10 roots in total. – Henning Makholm Jan 16 '17 at 10:27

Both statements $(A)$&$(B)$ derive from counting multiplicity as stated in the exercise.
The statements $(C)$&$(D)$ might or might not be true, but there is no way to derive that from counting multiplicity.