Answer
$x=-2$ is a lower bound to the zeros of $f$
$x=3$ 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 $f$.
$f(x)=x^3-x^2-4x+2,$
$\begin{array}{lllll}
\underline{-2}| & 1 & -1&-4 &2\\
& & -2& 6& -4\\
& -- & -- & -- & --&--\\
& 1&-3 & 2 & -2
\end{array}$
Therefore, $x=-2$ is a lower bound to the zeros of $f$
$\begin{array}{lllll}
\underline{3}| & 1&-1 & -4 & 2\\
& & 3& 6 & 6\\
& -- & -- & -- & --&--\\
& 1&2& 2 & 8
\end{array}$
Therefore, $x=3$ is an upper bound to the zeros of $f$.
