How can I factor the function f(x) = x^4 – 64x^2 into linear factors?

To factor the function f(x) = x⁴ – 64x², we can start by noting that it has a common factor of x². First, we can factor out x² from the expression:

f(x) = x²(x² – 64)

Next, we see that the remaining quadratic, x² – 64, is a difference of squares, which can be factored as:

x² – 64 = (x – 8)(x + 8)

Now, substituting this back into our equation, we have:

f(x) = x²(x – 8)(x + 8)

At this point, we can also further factor x² into linear factors:

x² = x imes x

Thus, the complete factorization of f(x) into linear factors is:

f(x) = x imes x imes (x – 8) imes (x + 8)

In summary, the linear factorization of the function f(x) = x⁴ – 64x² is:

f(x) = x(x)(x – 8)(x + 8)

This shows the function written as a product of its linear factors.