Question

Factor the following polynomial function completely. use the conjugate roots theorem, if applicable. f(x)=x⁶-13x⁴-52x²-64

Answer 1

To factor the polynomial function f(x) = x⁶ - 13x⁴ - 52x² - 64 completely, use the conjugate roots theorem. The polynomial can be factored as (x² - 4)(x² + 4)(x² + 8).

Step-by-step explanation:

To factor the polynomial function f(x) = x⁶ - 13x⁴ - 52x² - 64 completely, we can use the conjugate roots theorem. Since the polynomial has only even powers of x, we can rewrite it as:

f(x) = (x²)³ - 13(x²)² - 52x² - 64

Now, we can substitute x² with a variable, let's say y.

So, we have f(y) = y³ - 13y² - 52y - 64.

By factoring the polynomial f(y) using synthetic division or long division, we find that it can be factored as f(y) = (y - 4)(y + 4)(y + 8).

Therefore, the original polynomial function f(x) can be factored as f(x) = (x² - 4)(x² + 4)(x² + 8).