Q.Find a polynomial equation of minimum degree with rational coefficients having $i-2$ as a root.

Uses the Conjugate Root Theorem — for a polynomial with rational coefficients, complex roots occur in conjugate pairs — to find the second root, then multiplies the two linear factors.

1. Note "$i-2$" is the complex number $-2+i$. This is the given root.
2. Conjugate Root Theorem: if a polynomial has rational (in particular real) coefficients and $a+ib$ ($b\ne0$) is a root, then its conjugate $a-ib$ must also be a root.
3. So the other root is $\overline{-2+i}=-2-i$.
4. The minimum-degree polynomial with exactly these two roots is $(x-(-2+i))(x-(-2-i))=(x+2-i)(x+2+i)$. …