Question

list the possible zeros. choose all that apply. \\( f(x) = 2x^3 + 7x^2 - 28x + 30 \\) \\( pm \frac{1}{10} \\) \\( pm \frac{1}{4} \\) \\( pm \frac{2}{3} \\) \\( pm \frac{1}{2} \\) \\( pm 5 \\) \\( pm 7 \\) \\( pm 30 \\) \\( pm \frac{3}{2} \\) \\( pm 3 \\) \\( pm 6 \\) \\( pm 10 \\) \\( pm 1 \\) \\( pm 28 \\) \\( pm \frac{5}{2} \\) \\( pm \frac{2}{15} \\) \\( pm 15 \\) \\( pm \frac{1}{20} \\) \\( pm \frac{15}{2} \\) \\( pm \frac{2}{5} \\) there are no possible zeros \\( pm 2 \\)

Explanation:

Step1: List constant term factors

Factors of $30$: $\pm1, \pm2, \pm3, \pm5, \pm6, \pm10, \pm15, \pm30$

Step2: List leading coefficient factors

Factors of $2$: $\pm1, \pm2$

Step3: Apply Rational Root Theorem

Possible zeros: $\pm\frac{1}{1}, \pm\frac{1}{2}, \pm\frac{2}{1}, \pm\frac{2}{2}, \pm\frac{3}{1}, \pm\frac{3}{2}, \pm\frac{5}{1}, \pm\frac{5}{2}, \pm\frac{6}{1}, \pm\frac{6}{2}, \pm\frac{10}{1}, \pm\frac{10}{2}, \pm\frac{15}{1}, \pm\frac{15}{2}, \pm\frac{30}{1}, \pm\frac{30}{2}$

Step4: Simplify duplicates

Simplify to get: $\pm 1$, $\pm 2$, $\pm 3$, $\pm 5$, $\pm 6$, $\pm 10$, $\pm 15$, $\pm 30$, $\pm \frac{1}{2}$, $\pm \frac{3}{2}$, $\pm \frac{5}{2}$, $\pm \frac{15}{2}$

Answer:

$\pm 1$, $\pm 2$, $\pm 3$, $\pm 5$, $\pm 6$, $\pm 10$, $\pm 15$, $\pm 30$, $\pm \frac{1}{2}$, $\pm \frac{3}{2}$, $\pm \frac{5}{2}$, $\pm \frac{15}{2}$