To perform synthetic division of the polynomial \(3x^3 - 10x^2 + 5x + 2\) by \(x - 1\), we can follow these steps:
- Write down the coefficients of the polynomial: \(3, -10, 5, 2\).
- Since we are dividing by \(x - 1\), we use \(1\) for synthetic division.
- Set up synthetic division by writing \(1\) on the left and the coefficients on the right:
1 | 3 -10 5 2
|
- Bring down the leading coefficient to the bottom row:
1 | 3 -10 5 2
|
| 3
- 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
- Add the numbers in the second column: \(-10 + 3 = -7\).
1 | 3 -10 5 2
| 3 -7
| 3 -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
- Add \(5 + (-7) = -2\):
1 | 3 -10 5 2
| 3 -7 -2
| 3 -7 -2
- Now multiply \(-2\) by \(1\) and write the result under \(2\):
1 | 3 -10 5 2
| 3 -7 -2
| 3 -7 -2
- 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:
\[ 3x^2 - 7x - 2 \quad \text{with no remainder.} \]