Question

use the theorem on bounds to establish the best integral bounds for the roots of the following equation. 2x^3 - 4x^2 + 344 = 0. the best integral bounds are <x<.

Explanation:

Step1: Divide by leading - coefficient

Divide the polynomial $2x^{3}-4x^{2}+344 = 0$ by the leading - coefficient 2 to get $x^{3}-2x^{2}+172 = 0$. Let $f(x)=x^{3}-2x^{2}+172$.

Step2: Apply the upper - bound rule

For the upper - bound, we use synthetic division. Start testing positive integers. When we test $x = 6$:
\[

$$\begin{array}{c|ccc} 6&1&- 2&0&172\\ & &6&24&144\\ \hline &1&4&24&316 \end{array}$$

\]
Since all the numbers in the bottom row are non - negative, 6 is an upper bound.

Step3: Apply the lower - bound rule

For the lower - bound, we test negative integers. When we test $x=-6$:
\[

$$\begin{array}{c|ccc} -6&1&-2&0&172\\ & & - 6&48&-288\\ \hline &1&-8&48& - 116 \end{array}$$

\]
When we test $x=-5$:
\[

$$\begin{array}{c|ccc} -5&1&-2&0&172\\ & &-5&35&-175\\ \hline &1&-7&35& - 3 \end{array}$$

\]
When we test $x=-4$:
\[

$$\begin{array}{c|ccc} -4&1&-2&0&172\\ & &-4&24&-96\\ \hline &1&-6&24&76 \end{array}$$

\]
Since the numbers in the bottom row alternate in sign when we use $x=-4$, - 4 is a lower bound.

Answer:

$-4