Question

Write a linear factorization of the function.
f(x) = x4 + 16x2

Answer 1

f(x) = x^2*(x - 4 i)*(x + 4 i)

Explanation:

Given

f(x) = x^4 + 16*x^2

First, take the Greatest Common Factor, as follows

f(x) = x^2*(x^2 + 16)

Then, find the roots of (x^2 + 16), as follows

x^2 + 16 = 0

x^2 = -16

x = sqrt(-16)

x = sqrt(-1*16)

x = sqrt(-1)*sqrt(16)

There are 2 solutions, x1 = 4 i and x2 = -4 i; both of them imaginary. Then, the roots are (x - 4 i) and (x + 4 i). Replacing in the second equation:

f(x) = x^2*(x - 4 i)*(x + 4 i)

And that is the linear factorization.

Answer 2

f ( x ) = x^4 + 16 x² =
= x² · ( x² + 16 )
For x² + 16:
x 1/2 = (0+/-√(-64))/2= +/-8 i /2 = +/- 4 i
x² + 16 = ( x + 4 i ) · ( x - 4 i )
Answer:
f ( x ) = x² · ( x + 4 i ) · ( x - 4 i )