Question

Factor the following polynomial function completely. use the conjugate roots theorem, if applicable. f(x)=x⁶-6x⁴-31x² 36

Answer 1

To factor the polynomial function f(x) = x⁶ - 6x⁴ - 31x² + 36 completely, you can use the conjugate roots theorem. The completely factored form of the polynomial function is f(x) = (x² - 6)(x² - 4)(x² - 9).

Step-by-step explanation:

To factor the polynomial function f(x) = x⁶ - 6x⁴ - 31x² + 36 completely, we can use the conjugate roots theorem. First, let's group the terms with common factors:

x⁶ - 6x⁴ - 31x² + 36 = (x⁶ - 6x⁴) - (31x² - 36)

Now, we can factor out the common factors:

x⁴(x² - 6) - (31x² - 36)

Next, let's simplify the expression:

x⁴(x² - 6) - (x² - 6)(31x² - 36)

Now we have a quadratic expression that can be factored further:

(x² - 6)(x⁴ - 31x² + 36)

Finally, we can factor the second term using the quadratic formula or by factoring:

(x² - 6)(x² - 4)(x² - 9)

Therefore, the completely factored form of the polynomial function is f(x) = (x² - 6)(x² - 4)(x² - 9).