Final answer:
To factorize the function f(x) = x⁴ + 36x², we set it equal to zero and solve for x. The linear factorization is f(x) = x²(x² + 36).
Step-by-step explanation:
The given function is f(x) = x⁴ + 36x². To factorize this function, we need to find its roots. Let's set f(x) equal to zero and solve for x: x⁴ + 36x² = 0. Factoring out x², we get x²(x² + 36) = 0. Now, we set each factor equal to zero and solve for x: x² = 0 and x² + 36 = 0. The first equation gives us x = 0, and the second equation has no real solutions because a square plus a positive number can never be equal to zero. Therefore, the linear factorization of the function f(x) = x⁴ + 36x² is f(x) = x²(x² + 36).