note: Enter your answer and show all the steps that you use to solve this problem in the space provided.What is the quotient in simplest form?%20State%20any%20restrictions%20on%20the%20variable.z2%E2%88%924z%E2%88%923%C3%B7z+2z2+z%E2%88%9212
To find the quotient in simplest form, we first need to perform polynomial long division:
```
________________________
z + 2z^2 + z - 12 | z^2 - 4z - 3
```
Since `z^2` is the highest degree term, we divide `z^2` by `2z^2` and get `1/2`. We then multiply `(1/2)(z + 2z^2 + z - 12)` to get `(1/2)z + z + 1/2` and subtract it from the dividend:
```
________________________
z + 2z^2 + z - 12 | z^2 - 4z - 3
- (1/2)z - z - 1/2
_______________________
1/2z - 1/2
```
Our new dividend is `1/2z - 1/2`. Since the highest degree term is `z` in `1/2z - 1/2`, we divide `z` by `z` and get `1/2`. We then multiply `(1/2)(z + 2z^2 + z - 12)` to get `(1/2) + 1 + 1/2` and subtract it from the dividend:
```
________________________
z + 2z^2 + z - 12 | z^2 - 4z - 3
- (1/2)z - z - 1/2
_______________________
1/2z - 1/2
- 1/2z - z/2 - 1/2
______________________
1/2z + 1/2
```
Since the highest degree term is `1/2z` in `1/2z + 1/2`, we divide `1/2z` by `z` and get `1/2`. We then multiply `(1/2)(z + 2z^2 + z - 12)` to get `1/2 + 1 + 1/2` and subtract it from the dividend:
```
________________________
z + 2z^2 + z - 12 | z^2 - 4z - 3
- (1/2)z - z - 1/2
_______________________
1/2z - 1/2
- 1/2z - z/2 - 1/2
______________________
1/2z + 1/2
- 1/2z - 1/2
__________________
1
```
Therefore, the quotient is `1` and there are no restrictions on the variable `z`.