find the value of $a\in\Bbb{Z}$ such that $2+\sqrt{3}$ is a root of the polynomial?

Question

Find the value of $a\in\Bbb{Z}$ such that $2+\sqrt{3}$ is a root of the polynomial

$$x^3-5x^2 +ax -1$$ I got the answer and value of $a = -10 -5\sqrt{3} -1$.

Is my answer is correct or not?

Answer 1

We need $$(2+\sqrt3)^3-5(2+\sqrt3)^2+a(2+\sqrt3)-1=0,$$ which gives $$a=2-\sqrt3-(2+\sqrt3)^2+5(2+\sqrt3)=5$$

Answer 2

If $2+\sqrt3$ is a root, then so is $2-\sqrt3$

If $x$ is the third root, then by Vieta's theorem we have:

$$(2+\sqrt3)*(2-\sqrt3)*x=1$$ as $1$ is the free coefficient

Hence $x=1$

Then again, by Vieta's theorem:

$$a=(2+\sqrt3)*1+(2-\sqrt3)*1+(2+\sqrt3)*(2-\sqrt3)=5$$

Answer 3

If $a \in \mathbb Z$, then if $2+\sqrt3$ is a root, then so is $2-\sqrt3$.

Therefore, $x^3-5x^2 +ax -1$ is divisible by $x^2 - 4 x + 1$, which has $2\pm\sqrt3$ as roots.

Now, $x^3-5x^2 +ax -1 = (x^2 - 4 x + 1)(x-1)+(a-5)x$ and so $a=5$.