Question

fully factor (f(x) = x^3 - 10x^2 + 30x - 24), including factors for irrational zeros. use radicals, not decimal approximations.

(f(x) =)

Explanation:

Step1: Find a rational zero

Test divisors of \(-24\) to find a rational root:

$$f(6) = 6^3 - 10(6)^2 + 30(6) - 24 = 216 - 360 + 180 - 24 = 0$$

Step2: Perform polynomial division

Divide \(f(x)\) by \((x - 6)\) to find the quotient:

$$\frac{x^3 - 10x^2 + 30x - 24}{x - 6} = x^2 - 4x + 4$$

Step3: Solve the quadratic equation

Set the quotient to zero and solve for \(x\):

$$x^2 - 4x + 4 = 0 \implies (x - 2)^2 - 2 = 0 \implies x = 2 \pm \sqrt{2}$$

Step4: Write the fully factored form

Express \(f(x)\) using all found linear factors:

$$f(x) = (x - 6)(x - (2 + \sqrt{2}))(x - (2 - \sqrt{2}))$$

Answer:

\(f(x) = (x - 6)(x - 2 - \sqrt{2})(x - 2 + \sqrt{2})\)