Answer
$1, -1$ and $\frac{5}{2}$
Work Step by Step
We are given the function:
$$f(x)=6x^3-10x^2-6x+10.$$
We use a spreadsheet to determine the value of $f(x)$ for several values of $x$:
\[ \begin{array}{|c|c|c|}
\hline
& A & B \\
\hline
1 &x & f(x)\\
\hline
2 & 0 & 10\\
\hline
3 & 1 & 0\\
\hline
4 & 2 & 6\\
\hline
5 & 3 & 64\\
\hline
6 & 4 & 210\\
\hline
7 & 5 & 480\\
\hline
\end{array}\]
The spreadsheet shows that $f(1)=0$, which means $1$ is a zero of $f$. Apply synthetic division and we find that $f$ can be factored as:
$$\begin{align*}
f(x)&=(x-1)(6x^2-4x-10)\\
&=2(x-1)(3x^2+3x-5x-5)\\
&=2(x-1)(2x(x+1)-5(x+1))\\
&=2(x-1)(x+1)(2x-5).
\end{align*}$$
This means the zeroes of the function are $1, -1$ and $\frac{5}{2}$.
