Question

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)

Answer 1

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.

Answer 2

D

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.