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