Question

Use synthetic division to find the result when 3, x, cubed, minus, 10, x, squared, plus, 5, x, plus, 23x
3
−10x
2
+5x+2 is divided by x, minus, 1x−1.

Answer 1

To perform synthetic division of the polynomial \(3x^3 - 10x^2 + 5x + 2\) by \(x - 1\), we can follow these steps:

1. Write down the coefficients of the polynomial: \(3, -10, 5, 2\).
2. Since we are dividing by \(x - 1\), we use \(1\) for synthetic division.
3. Set up synthetic division by writing \(1\) on the left and the coefficients on the right:

```
1 | 3 -10 5 2
|
```

4. Bring down the leading coefficient to the bottom row:

```
1 | 3 -10 5 2
|
| 3
```

5. Multiply the value just brought down \(3\) by \(1\) (the number we are dividing by) and write the result under the next coefficient:

```
1 | 3 -10 5 2
| 3
| 3 -7
```

6. Add the numbers in the second column: \(-10 + 3 = -7\).

```
1 | 3 -10 5 2
| 3 -7
| 3 -7
```

7. Repeat the process: multiply \(-7\) by \(1\) and write under the next coefficient \(5\):

```
1 | 3 -10 5 2
| 3 -7
| 3 -7 -2
```

8. Add \(5 + (-7) = -2\):

```
1 | 3 -10 5 2
| 3 -7 -2
| 3 -7 -2
```

9. Now multiply \(-2\) by \(1\) and write the result under \(2\):

```
1 | 3 -10 5 2
| 3 -7 -2
| 3 -7 -2
```

10. Add the last column: \(2 + (-2) = 0\).

The final row gives you the coefficients of the resulting polynomial and the remainder.

The result of the synthetic division is:

\[
3x^2 - 7x - 2
\]

with a remainder of \(0\).

Thus, \(3x^3 - 10x^2 + 5x + 2\) divided by \(x - 1\) is:

\[
3x^2 - 7x - 2 \quad \text{with no remainder.}
\]