What is the linear factorization of the function f(x) = x^4 – 9x^2?

To factor the function f(x) = x⁴ – 9x², we start by observing that we can factor out a common term. Here, we notice that both terms contain x².

First, let’s rewrite the function:

f(x) = x²(x² – 9)

Next, we see that the expression in the parentheses, x² – 9, is a difference of squares. The difference of squares can be factored as follows:

a² – b² = (a – b)(a + b)

In our case, a is x and b is 3, since 9 = 3². Therefore, we can factor x² – 9 as:

x² – 9 = (x – 3)(x + 3)

Putting it all together, we have:

f(x) = x²(x – 3)(x + 3)

This is the linear factorization of the function. To summarize, the complete factorization of the given function is:

f(x) = x²(x – 3)(x + 3)