Explanation:
To find the remainder when 3x^3 - x^2 + 2x - 4 is divided by (x - 2), we can use synthetic division.
First, we write the coefficients of the polynomial in descending order of powers of x, with any missing terms represented by a coefficient of 0:
3 1 -1 2 -4
To perform synthetic division, we bring down the first coefficient (1) and multiply it by the divisor (x - 2) to get the first entry in the second row, which we then add to the second coefficient:
3 1 -1 2 -4
1
1
We repeat the process with the new coefficient in the third row, multiplying it by the divisor and adding it to the next coefficient:
3 1 -1 2 -4
1 3
1 2
Finally, we repeat the process with the new coefficient in the third row to get the last entry in the second row:
3 1 -1 2 -4
1 3 8
1 2 4
The last entry in the third row is the remainder, which is 4. Therefore, the remainder when 3x^3 - x^2 + 2x - 4 is divided by (x - 2) is 4.