Question

Which of the following is equivalent to the polynomial below?
x^2-8x+19

A.) (x+(4-3i)) (x+(4-3i))
B.) (x+(4+3i)) (x+(4-3i))
C.) (x-(4+3i)) (x-(4-3i))
D.) (x+(4+3i)) (x-(4-3i))

Answer 1

Answer:

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

Step-by-step explanation:

First, find the roots of the polynomial $x^2-8x+19:$

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

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

So,

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

Now the polynomial $x^2-8x+19$ is equivelent to

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