# Polynomials with complex coefficients

I am reading a book called College Algebra and it defines polynomials as an expression of the form $a_nx^n + a_{n-1}x^{n-1} + ...+ a_2x^2 + a_1x + a_0$ where $a_j$ are real numbers ( I'll stop the definition there). Does polynomials only have real coefficients or it can be complex numbers?

Polynomials can also have complex coefficients. There is actually quite a bit of work related to the topic. See, for example, here or here.

An example of such a polynomial would be

$$(42+42i)x^2+(7+i)x.$$

The roots of the above polynomial are at $$x=0$$ and $$x = -\frac{2}{21} + \frac{i}{14}.$$

So, nobody hinders you to create polynomials with complex coefficients.

To be clear, it's better to state over which ring the polynomial is defined.

A polynomial in $X$ over a ring $R$ is defined as $\sum\limits_{k = 0}^n = a_kX^K$, where $a_k \in R$ for $k \in \{0,\dots,n\}$.

In particular, a polynomial over $\Bbb{R}$ admits only real coefficients, whereas a polynomial over $\Bbb{C}$ admits complex coefficients.

Remarks: For pedagogical reasons, when one find (of ) in , one usually focus on real roots and discard the complex ones, especially in exercises of . I guess it's a possible reason that your College Algebra book only consider polynomials with real coefficients. Things will become clearer when you move on to .

• Is there a name for an analogue sum when the coefficients can be elements of a vector space (and $X$ a scalar) ? – Yves Daoust Apr 27 '18 at 13:58
• @YvesDaoust I don't have an idea. Say, a finite-dimensional $\Bbb{R}^n$, but I'm not sure how to make sense of multiplication. If it's the ring of matrices, surely that makes sense, but still I don't know whether there's a special name for that. – GNUSupporter 8964民主女神 地下教會 Apr 27 '18 at 14:03
• Just scalar multiplication. – Yves Daoust Apr 27 '18 at 14:05
• @YvesDaoust Sorry I don't know a special name for that. – GNUSupporter 8964民主女神 地下教會 Apr 27 '18 at 14:09