Here's an example.
Find the remainder using synthetic division:
(6x5-2x3+5x2-3x+7)/(x-2)
Rewrite the dividend as 6x5+0x4-2x3+5x2-3x+7. This ensures that we have all the powers of x in descending order.
Instead of dividing by x-2 we treat 2 as the divisor (note that -2 becomes 2).
Now we set out the division using only the coefficients (no x's or powers of x):
2 | 6 0 -2 5 -3 | 7
6
6
The first coefficient is written as the first digit in the next two lines of the calculation. There are two vertical bars: one separates the divisor from the dividend and the other comes before the constant term. This bar will appear on the lines below as the calculation progresses and will separate the quotient from the remainder.
The underscore will be extended along the same line as the calculation proceeds.
We take the number in the first position in the borrom line and multiply it by the divisor so we have 6×2=12: This result is placed as the next number on the second line. Add this to the number directly above and put the result as the second number on the bottom line, 0+12=12
2 | 6 0 -2 5 -3 | 7
6 12
6 12
Repeat the process for the next column. 12×2=24. Write this number to follow the 12 then add to the number above -2+24=22 to follow 12 in the bottom line:
2 | 6 0 -2 5 -3 | 7
6 12 24
6 12 22
Repeat for the remaining numbers in the dividend:
2 | 6 0 -2 5 -3 | 7
6 12 24 44 98 | 190
6 12 22 49 95 | 197
The remainder is 197.