Question

Which of the following is equivalent to the polynomial below?

Answer 1

Answer:

$(x-(4+\sqrt{3}i))(x-(4-\sqrt{3}i))$

Step-by-step explanation:

Consider the quadratic expression

$x^2-8x+19$

First, find its discriminant:

$D=(-8)^2-4\cdot 1\cdot 19=64-76=-12$

Note that $i^2=-1,$ then $D=12i^2.$

Use quadratic formula to find the roots:

$x_{1,2}=\dfrac{-(-8)\pm \sqrt{12i^2}}{2\cdot 1}=\dfrac{8\pm 2\sqrt{3} i}{2}=4\pm \sqrt{3}i$

Now, given quadratic expression is equivalent to

$(x-(4+\sqrt{3}i))(x-(4-\sqrt{3}i))\\ \\=(x-4-\sqrt{3}i)(x-4+\sqrt{3}i)$