Correct answer only ! factor the polynomial function over the complex numbers.

(x + 1)(x - 2)(x + i√2)(x - i√2)

Step-by-step explanation:

f(x) = x⁴ - x³ - 2x - 4

possible rational roots: +/- 1, 2, 4

Use Synthetic Division to find two of the roots (so you are left with a quadratic polynomial)

-1  |   1     -1     0     -2     -4

   |   ↓     -1     2     -2      4

       1      -2    2     -4     0 → remainder is 0 so x = -1 is a root

                                        ⇒ (x + 1) is a factor

Now find another root using the decomposed polynomial:

2  |   1     -2     2     -4    

   |   ↓     2      0     4

       1      0     2     0 →remainder is 0 so x = 2 is a root

                               ⇒ (x - 2) is a factor

Now factor the decomposed polynomial:

      x² + 0x + 2

=    x² -  (-2)

=     +/-

=    (x + i√2)(x - i√2)

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