Question

The polynomial equation of degree $4$ having real coefficients with three of its roots as $2 \pm \sqrt{3}$ and $1+2i$. is

• A. $x^4 - 6x^3 - 14x^2 + 22x + 5 = 0$
• B. $x^4 - 6x^3 - 19x + 22x - 5 = 0$
• C. $x^4 - 6x^3 - 19x - 22x + 5 = 0$
• D. $x^4 - 6x^3 + 14x^2 -22x + 5 = 0$

Answer 1

Answer: (D)

$x^4 - 6x^3 + 14x^2 -22x + 5 = 0$

Answer 2

It is given that, the polynomial equation of degree 4 having real coefficients with three of its roots as $2 \pm \sqrt{3}$ and $1+2 i$, so the remaining root is $1-2 i$.
Now, the quadratic equation whose roots as $2 \pm \sqrt{3}$ is
$x^{2}-4 x+1=0,$ and
the quadratic equation whose roots as $1 \pm 2 i$, is
$x^{2}-2 x+5=0$
So, the required polynomial equation is
$\left(x^{2}-4 x+1\right)\left(x^{2}-2 x+5\right)=0$
$\Rightarrow x^{4}-6 x^{3}+14 x^{2}-22 x+5=0$