Final answer:
The function f(x) = x^4 + 9x^2 can be linearly factored by treating it as a sum of squares with complex numbers, leading to the factorization f(x) = (x^2 + 3i)(x^2 - 3i).
Step-by-step explanation:
To perform the linear factorization of the function f(x) = x4 + 9x2, we first notice that this is a quadratic in terms of x2, which can be factored as a sum of squares. We can rewrite the equation as f(x) = (x2)2 + (32)(x2). This is not factorable as a sum of squares in real numbers, but we can use complex numbers to further factor the equation.
We let u = x2 and consider f(u) = u2 + 9. This is a sum of squares and can be factored using complex numbers as u2 + (3i)2 = (u + 3i)(u - 3i).
Substituting x2 back in for u, we have (x2 + 3i)(x2 - 3i). Since we can't further factor x2 + 3i or x2 - 3i over the real numbers, our final linear factorization, using complex numbers, is f(x) = (x2 + 3i)(x2 - 3i).