Question

If $f\left(x\right)$ is a polynomial of degree $n$ with rational coefficients, and $1+2i,2-\sqrt{3}$ and $5$ are three roots of $f\left(x\right)=0$, then the least value of $n$ is

• A. $5$
• B. $4$
• C. $0.5$
• D. $6$

Answer 1

The correct option is A $5$

The coefficients are rational, the roots should be either rational or if irrational or complex, must occur in pairs with their conjugates.

$\because 1+2i,2-\sqrt{3}$ and $5$ are the three roots given.
The irrational root $2-\sqrt{3}$ will have its conjugate $2+\sqrt{3}$ a root as well.
Similarly, $1-2i$ is also a root.
$\therefore$ there exist at least $5$ roots of $f\left(x\right)=0$ and hence least value of $n$ is $5$.