Question

use the rational zero theorem to create a list of all possible rational zeroes of the function $f(x) = 14x^7 - 4x^2 + 2$.(1 point)\\(\boldsymbol{\pm1,\pm2,\pm\frac{1}{2},\pm\frac{1}{7},\pm\frac{1}{14},\pm\frac{2}{7}}\\)\\(\pm1,\pm2,\pm7,\pm14,\pm\frac{7}{2}\\)\\(\pm1,\pm2,\pm7,\pm14\\)\\(\pm1,\pm2,\pm\frac{1}{7},\pm\frac{1}{14}\\)

Explanation:

Step1: Identify constant term factors

The constant term is 2. Its factors are $\pm1, \pm2$.

Step2: Identify leading coefficient factors

The leading coefficient is 14. Its factors are $\pm1, \pm2, \pm7, \pm14$.

Step3: Apply Rational Zero Theorem

Possible rational zeros are $\pm\frac{\text{Factors of 2}}{\text{Factors of 14}}$. Calculate all combinations:
$\pm\frac{1}{1}=\pm1$, $\pm\frac{1}{2}$, $\pm\frac{1}{7}$, $\pm\frac{1}{14}$, $\pm\frac{2}{1}=\pm2$, $\pm\frac{2}{7}$

Step4: Compile the full list

Combine all unique values: $\pm1, \pm2, \pm\frac{1}{2}, \pm\frac{1}{7}, \pm\frac{1}{14}, \pm\frac{2}{7}$

Answer:

$\pm1, \pm2, \pm\frac{1}{2}, \pm\frac{1}{7}, \pm\frac{1}{14}, \pm\frac{2}{7}$