Question

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

Answer 1

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

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)