Answer
$x=4$ is a lower bound to the zeros of $f$
$x=5$ is an upper bound to the zeros of $f$.
Work Step by Step
Bounds on Zeros
Let $f$ denote a polynomial function whose leading coefficient is positive.
• If $M > 0$ is a real number and if the third row in the process of synthetic
division of $f$ by $x - M$ contains only numbers that are positive or zero,
then $M$ is an upper bound to the zeros of $f$.
• If $m < 0$ is a real number and if the third row in the process of synthetic
division of $f$ by $x - m$ contains numbers that alternate positive (or 0) and
negative (or 0), then $m$ is a lower bound to the zeros of $f$.
$f(x)=2x^3-7x^2-10x+35,$
$\begin{array}{lllll}
\underline{-3}| & 2 & -7&-10&35\\
& & -6& 39& -87\\
& -- & -- & -- & --&--\\
& 2&-13 & 29 & -52
\end{array}$
Therefore, $x=-3$ is a lower bound to the zeros of $f$
$\begin{array}{lllll}
\underline{5}| & 2&-7 & -10 & 35\\
& & 10& 15 & 25\\
& -- & -- & -- & --&--\\
& 2&3& 5 & 60
\end{array}$
Therefore, $x=5$ is an upper bound to the zeros of $f$.
