Which of the following options correctly represents the complete factored form of the polynomial F (x) = x^4-3x^2-4?

A. F (x) = (x+1) (x-1) (x+2) (x-2)

B. F (x) = (x+i) (x-i) (x+2i) (x-2i)

C. F (x) = (x+1) (x-1) (x+2i) (x-2i)

D. F (x) = (x+i) (x-i) (x+2) (x-2)

D

Step-by-step explanation:

Let's substitute a for x²:

x^4 - 3x² - 4

a² - 3a - 4

Now, this looks like something that is much more factorisable:

a² - 3a - 4 = (a - 4) (a + 1)

Plug x² back in for a:

(a - 4) (a + 1)

(x² - 4) (x² + 1)

The first one is a difference of squares, which can be factored into:

x² - 4 = (x + 2) (x - 2)

The second one can also be treated as a difference of squares:

x² + 1 = x² - (-1) = (x + √-1) (x - √-1) = (x + i) (x - i)

The answer is (x + 2) (x - 2) (x + i) (x - i), or D.

F (x) = x⁴-3x²-4

x⁴-3x²-4=

=x⁴-4x² + x²-4

=x² (x²-4) + (x²-4)

= (x²+1) (x²-4)

= (x+i) (x-i) (x+2) (x-2)

F (x) = (x+i) (x-i) (x+2) (x-2) D.