Write a linear factorization of the function.

f (x) = x4 + 49x2

options:

f (x) = x2 (7x + i) 2

f (x) = x2 (x + 7i) (x - 7i)

f (x) = x2 (7x + i) (7x - i)

f (x) = x2 (x + 7i) 2

f (x) = x^2 (x + 7i) (x - 7i)

Step-by-step explanation:

Factoring out x^2 gives ...

f (x) = x^2 (x^2 + 49)

The factor with 49 can be considered to be the difference of two squares, where one of the squares is - 49. Then its square root is ±7i, and the factorization of that term is ...

x^2 + 49 = (x + 7i) (x - 7i)

So, the overall factorization of f (x) is ...

f (x) = x^2 (x + 7i) (x - 7i)