Answer:
Option 3
[tex]f(x) = x^4 - 35x^2+180x -416[/tex]
Step-by-step explanation:
To answer this question we start by writing the polynomial product form of factors:
[tex](x-4)(x + 8)(x- (2 + 3i))(x- (2-3i))[/tex]
We multiply the first two factors:
[tex](x^2 + 8x -4x -32)(x- (2 + 3i))(x- (2-3i))\\\\(x^2 + 4x -32)(x- (2 + 3i))(x- (2-3i))[/tex]
Now we multiply the second two factors:
[tex](x^2+ 4x -32) (x^2 -2x+ 3ix -2x -3ix + 4 - 9i^2)[/tex]
We know that [tex]i = \sqrt{(-1)}[/tex]
So:
[tex]i^2 = -1[/tex]
[tex](x^2 + 4x -32) (x^2 -4x + 4 + 9)\\\\(x^2 + 4x -32) (x^2 -4x + 13)[/tex]
Finally we multiply both terms and obtain the polynomial sought:
[tex](x^4 -4x^3 + 13x^2 + 4x^3 -16x^2 +52x-32x^2+128x -416)\\\\x^4 - 35x^2+180x -416[/tex]
Finally the correct option is the third.
[tex]f(x) = x^4 - 35x^2+180x -416[/tex]
