Chapter 4.5, Problem 27E

To determine

To find: An integral upper bound using Upper Bound Theorem and to find an integral lower bound of the zeros using Lower Bound Theorem, of $f\left(x\right)=x^2-x-1$

Answer to Problem 27E

Upper bound: 2

Lower bound: $-1$

Explanation of Solution

Given information: $f\left(x\right)=x^2-x-1$

Calculation:

$f\left(x\right)=x^2-x-1$
r	1 - 1 - 1
1	1 0	- 1
2	1 1	1

Since all quotient values for 2 are positive,

  $x=2\text{ is}$ an upper bound

$f\left(x\right)=x^2-x-1$
r	11 - 1
1	1 2	1

Since all quotient values of $f\left(-x\right)$ are positive,

  $x=-1\text{ is}$ a lower bound.

Upper bound: 2

Lower bound: $-1$