Find all zeros of the function and write the polynomial as a

Find all zeros of the function and write the polynomial as a product of linear factors.
f(x) = 3x⁴ + 4x³ + 13x² + 16x + 4

a. f(x) = (3x - 1)(x - 1)(x + 2)(x - 2)
b. f(x) = (3x + 1)(x + 1)(x + 2i)(x - 2i)
c. f(x) = (3x - 1)(x - 1)(x + 2i)(x - 2i)
d. f(x) = (3x + 1)(x + 1)(x + 2)(x - 2)

Solution

f(x) = 3x⁴ + 4x³ + 13x² + 16x + 4

Apply rational root theorem :

The factors of the leading coefficient (3) are 1 3 .The factors of the constant term (4) are 1 2 2 4 . Then the Rational Roots Tests yields the following possible solutions:

±1/1, ±1/3, ±2/1, ±2/3, ±2/1, ±2/3, ±4/1, ±4/3

Substitute the POSSIBLE roots one by one into the polynomial to find the actual roots. Start first with the whole numbers.

If we plug these values into the polynomial P(x), we obtain P(1)=0.

(3x⁴ + 4x³ + 13x² + 16x + 4)/(x +1) = 3x3+x2+12x+4

Further apply rational root theorem to find remainig roots

Roots are x = -1 , -1/3 , 2i , -2i

f(x) = (x+1)(3x+1)(x+2i)(x-2i)

Option b