Question

A polynomial has one root that equals 2+I. Name one other root of this polynomial.

Answer 1

2-i. Complex roots always come in pairs (complex conjugates)

Answer 2

The other root is 2-i.

Explanation:

First, we need to make an assumption: the polynomials has real coefficients. Otherwise, the polynomials can be P(x) = x-2-i = x - (2+i), which has complex coefficients, and has only one root. Then, the next reasoning is completely valid assuming that the polynomials has real coefficients.

Polynomials with real coefficients can have complex roots, and the simplest example is P(x) = x²+1. Now, the interesting fact is that those complex roots come in conjugate pairs. Notice that i is a root of x²+1, but -i is also a root.

So, if 2+i is a root of certain polynomials, its conjugate is also a root of the same polynomial, i.e. , 2-i is also a root.