Question

use the theorem on bounds to establish the best integral bounds for the roots of the following equation. 2x^3 - 5x^2 + 141 = 0 the best integral bounds are -4 < x < 2.

Explanation:

Step1: Identify the polynomial coefficients

The polynomial is $2x^{3}-5x^{2}+0x + 141$, with coefficients $a_3 = 2$, $a_2=-5$, $a_1 = 0$, $a_0=141$.

Step2: Apply the upper - bound theorem

We use synthetic division. First, test positive integers. Start with $x = 2$.
When we perform synthetic division with 2:

2	2	-5	0	141
-	-	-	-	-
		4	- 2	- 4
	2	- 1	- 2	137

Since all the numbers in the bottom row (except the remainder) are non - negative when dividing by 2, 2 is an upper bound.

Step3: Apply the lower - bound theorem

Test negative integers. Start with $x=-4$.
When we perform synthetic division with - 4:

-4	2	-5	0	141
-	-	-	-	-
		- 8	52	- 208
	2	- 13	52	- 67

Since the numbers in the bottom row alternate in sign when dividing by - 4, - 4 is a lower bound.

Answer:

The best integral bounds are $-4