The other root is 2-i.
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.
One other root of this polynomial is
One property of the complex root of a polynomial is the conjugate root property.
If one root of a polynomial is
then the conjugate is also a root of this polynomial.
The conjugate of
Also the conjugate of
is a root of the polynomial, then its complex conjugate
is also a root of that polynomial.
The same applies to purely imaginary complex roots too.
is a root then its conjugate
is also a root.
the conjugate of 2+i is 2-i is also a root.
left be aligning the decimal points is required to not required see the number of digits to the right