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Question 1186858: Consider the following polynomial.
F(x) = x^3 + x^2 − 22x − 40
Use synthetic division to identify integer bounds of the real zeros. Find the least upper bound and the greatest lower bound guaranteed by the Upper and Lower.
Upper Bound:
Lower Bound:
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to find the integer bounds using synthetic division:
**Understanding Upper and Lower Bound Theorem**
The Upper and Lower Bound Theorem helps us find integer values that are greater than or less than all real zeros of a polynomial. If we perform synthetic division with a positive number *c* and all the numbers in the bottom row are positive or zero, then *c* is an *upper bound* for the real zeros. If we perform synthetic division with a negative number *c* and the numbers in the bottom row alternate in sign (positive, negative, positive, etc., or zero), then *c* is a *lower bound* for the real zeros.
**Applying Synthetic Division**
Let's test some integer values:
* **Testing x = 6:**
```
6 | 1 1 -22 -40
| 6 42 120
------------------
1 7 20 80
```
Since all the numbers in the bottom row are positive, 6 is an *upper bound*.
* **Testing x = -5:**
```
-5 | 1 1 -22 -40
| -5 20 10
------------------
1 -4 -2 -30
```
The signs don't alternate.
* **Testing x = -4:**
```
-4 | 1 1 -22 -40
| -4 12 40
------------------
1 -3 -10 0
```
-4 is a zero of the polynomial, which is acceptable.
* **Testing x = -3:**
```
-3 | 1 1 -22 -40
| -3 6 48
------------------
1 -2 -16 8
```
The signs don't alternate.
* **Testing x = -2:**
```
-2 | 1 1 -22 -40
| -2 2 40
------------------
1 -1 -20 0
```
-2 is also a zero of the polynomial.
* **Testing x = -1:**
```
-1 | 1 1 -22 -40
| -1 0 22
------------------
1 0 -22 -18
```
The signs don't alternate.
**Results:**
* **Upper Bound:** 6 (all positive or zero in bottom row)
* **Lower Bound:** -4 (alternating signs in bottom row, or a zero)
**Important Note:** The Upper and Lower Bound Theorem gives us *bounds* for the real zeros. It doesn't guarantee that the bounds themselves *are* zeros. In this case, -4 and -2 are zeros, however, that does not have to be the case.
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