# Is every real root also a complex root? (Complex zeros and Fundamental Theorem of Algebra)

Stupid question about this topic: I don't understand the rule that says "A poly with degree N will have N complex roots". eg: $P(X)=x^5 + x^3 - 1$ is a 5th degree polynomial function, so P(x) has exactly 5 complex zeros. Sorry, but how do you know it will have 5 complex zeros? Maybe it factors into only real/integer roots? Or, is every real root also a complex root? ie: $x = x + 0i$

What is a complex polynomial function (which has at least one complex zero)? Just a poly with an "i" in the coefficient? eg: $4ix^2 + ...$

Disclaimer: I am not a student trying to get free internet homework help. I am an adult who is learning Calculus from a textbook. I am deeply grateful to the members of this community for their time.

• Real roots are special complex roots, those with imaginary part $0$. The mention of "complex" only indicates that one doesn't have to leave $\mathbb{C}$ to find the zeros. Nov 4, 2013 at 22:23
• Your textbook should also say something about the roots being "counted according to their multiplicity". $(x - 1)^2(x - i)^2$ is a polynomial of degree 4 with just 2 distinct roots, $1$ and $i$. Nov 4, 2013 at 22:35

Yes, real numbers are complex numbers with imaginary part $0$.
The real numbers are a subset of the complex numbers ($\mathbb{R}\subset\mathbb{C}$), so any element of the reals is also complex.