Answer
The possible rational zeroes for the function \[f\left( x \right)={{x}^{5}}-{{x}^{4}}-7{{x}^{3}}+7{{x}^{2}}-12x-12\] are \[\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12\].
Work Step by Step
Here, the constant term is $-12$ and the leading coefficient is 1.
The factors of the constant term, $-12$ are $\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12$ and the factors of the leading coefficient, 1 are $\pm 1$.
So, the list of all possible rational zeroes is calculated by the formula:
$\begin{align}
& \text{Possible rational zeroes}=\frac{\text{Factors of the constant term}}{\text{Factors of the leading coefficient}} \\
& =\frac{\text{Factors of }-12}{\text{Factors of 1}} \\
& =\frac{\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12}{\pm 1} \\
& =\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12
\end{align}$
Therefore, there are total twelve possible rational zeroes for the function $f\left( x \right)={{x}^{5}}-{{x}^{4}}-7{{x}^{3}}+7{{x}^{2}}-12x-12$ that are $\pm 1,\pm 2,\pm 3,\pm 4,\pm 6,\pm 12$.
