Question 1186858
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.