Question

fully factor (f(x) = x^3 - x^2 - 4x - 2), including factors for irrational zeros. use radicals, not decimal approximations.

(f(x) = )

Explanation:

Step1: Find a rational root

Test \(x = -1\) using the Rational Root Theorem:

$$f(-1) = (-1)^3 - (-1)^2 - 4(-1) - 2 = -1 - 1 + 4 - 2 = 0$$

Step2: Divide by the factor

Divide \(f(x)\) by \((x + 1)\) using synthetic division:

$$f(x) = (x + 1)(x^2 - 2x - 2)$$

Step3: Find remaining roots

Solve \(x^2 - 2x - 2 = 0\) using the quadratic formula:

$$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-2)}}{2(1)} = 1 \pm \sqrt{3}$$

Step4: Write the fully factored form

Express \(f(x)\) as a product of linear factors:

$$f(x) = (x + 1)(x - (1 + \sqrt{3}))(x - (1 - \sqrt{3}))$$

Answer:

\(f(x) = (x + 1)(x - 1 - \sqrt{3})(x - 1 + \sqrt{3})\)