Are real coefficients also complex coefficients?

Question

Can $x^2 + x +1$ be called a polynomial with complex coefficients?

I know that all real numbers are complex numbers, so does this hold here as well?

Answer 1

We know that $$P(x) = x^2 + x +1 = 1\cdot x^2 + 1\cdot x + 1\cdot 1\;$$ is a polynomial with integer coefficients, and so it is also a polynomial with rational coefficients, just as it a polynomial with real coefficients.

In this same spirit, it is entirely valid to call it a polynomial with complex coefficients.

Answer 2

Yes it can since $\,\Bbb R\subset \Bbb C\;$ , so every real number is also a complex number (but not the other way around!)

Answer 3

Yes and No: yes since $\mathbb R\subset \mathbb C$ and no since in this case you are far from accurate, and you may miss some properties which are valid only in $ \mathbb R $ and not $ \mathbb C $: take the example of symmetric matrix which is diagonalizable if it's real.

Answer 4

The polynomial $x^2 + x +1$ can certainly be called a polynomial with complex coefficients, but moreover the idea of doing so has important mathematical applications. For example, when proving that a real symmetric matrix has a real eigenvalue, it is very convenient to extend the scalars to $\mathbb{C}$, find an eigenvalue over the complex domain, and then show that the eigenvalue found is actually real. Moreover, historically speaking, mathematicians realized the importance of complex numbers by noticing that in order to find the real roots of some cubic polynomials by Cardano's formula, one necessarily passes through the complex domain in computing them! I can provide historical references if you are interested.