Question

10x^(4)-91x^(3)+17x^(2)-75x+26-:x-9
find the quotient

Answer 1

To divide the polynomial \(10x^4 - 91x^3 + 17x^2 - 75x + 26\) by \(x - 9\), we can use synthetic division.

### Steps for Synthetic Division:

1. Write down the coefficients of the polynomial: \(10, -91, 17, -75, 26\).
2. Write 9 (the zero of \(x - 9\)) to the left.
3. Perform the synthetic division.

#### Synthetic Division Process:

```
9 | 10 -91 17 -75 26
| 90 -9 72 -27
---------------------------------------
10 -1 8 -3 -1
```

### Explanation of Steps:

1. Bring down the leading coefficient (10).
2. Multiply 9 by 10 and write the result (90) under the next coefficient (-91).
3. Add -91 + 90 to get -1.
4. Multiply 9 by -1 to get -9, write it under the next coefficient (17).
5. Add 17 + (-9) to get 8.
6. Multiply 9 by 8 to get 72, write it under the next coefficient (-75).
7. Add -75 + 72 to get -3.
8. Multiply 9 by -3 to get -27, write it under the last coefficient (26).
9. Add 26 + (-27) to get -1.

### Result:

The bottom row gives the coefficients of the quotient and the remainder:
- The quotient is \(10x^3 - x^2 + 8x - 3\).
- The remainder is \(-1\).

### Final Answer:

So, the result of dividing \(10x^4 - 91x^3 + 17x^2 - 75x + 26\) by \(x - 9\) is:

\[
10x^3 - x^2 + 8x - 3 - \frac{1}{x - 9}
\]

Or simply, the quotient is

\[
10x^3 - x^2 + 8x - 3
\] with a remainder of \(-1\).