Question

type the correct answer in each box. use numerals instead of words. if necessary, use / for the fraction bar(s). find the factors of function f, and use them to complete this statement. f(x) = 2x⁴ − x³ − 18x² + 9x from left to right, function f has zeros at x = □, x = □, x = □, and x = □.

Explanation:

Step1: Factor out common term

$f(x) = x(2x^3 - x^2 - 18x + 9)$

Step2: Factor cubic by grouping

Group terms: $2x^3 - x^2 - 18x + 9 = x^2(2x - 1) - 9(2x - 1)$
Factor: $=(x^2 - 9)(2x - 1)$

Step3: Factor difference of squares

$x^2 - 9 = (x - 3)(x + 3)$
Full factored form: $f(x) = x(x + 3)(x - 3)(2x - 1)$

Step4: Solve for zeros

Set each factor to 0:

1. $x = 0$
2. $x + 3 = 0 \implies x = -3$
3. $x - 3 = 0 \implies x = 3$
4. $2x - 1 = 0 \implies x = \frac{1}{2}$

Step5: Order zeros left to right

Sort: $-3, 0, \frac{1}{2}, 3$

Answer:

$x = -3$, $x = 0$, $x = \frac{1}{2}$, and $x = 3$