Synthetic Division

Use synthetic division to divide 3x⁴ - 8x² - 11x +1 by x - 2

Step 1: In the first row, list the number from the divisor and the coefficients of the divided in order of decreasing powers of x, inserting 0 in for missing powers of x.

2] 3  0  -8  -11  1

Step 2: Skip a line, draw a line, and draw a partial box under the line beneath the last coefficient in the first row. Bring the first coefficient of the dividend below the line.

2] 3  0  -8  -11  1

                           
   3

Step 3: Multiply the divisor constant, 2, and the number below the line, 3. Place the product, 6, under the dividend's next coefficient.

2] 3  0  -8  -11  1

        6                    
   3   6

Step 4: Add the numbers in that column and write the sum below the column under the line.

2] 3  0  -8  -11  1

        6                  

   3   6   

Step 5: Multiply the divisor and the last entry of the last row. Place the product under the dividends next coefficient.

2] 3  0  -8  -11  1

        6  12               

   3   6   

Step 6: Add the numbers in that column and write the sum below the column under the line.

2] 3  0  -8  -11  1

        6  12               

   3   6   4

Step 7: Repeat steps 5 and 6 until a number appears in the box under the last column.

2] 3  0  -8  -11  1

        6  12    8  -6        

   3   6   4    -3  [ -5]

-The last number in the third row, in this case -5, is the remainder. The other numbers are the coefficients of the quotient. Therefore the quotient is 3x³ + 6x² + 4x - 3

If the remainder is 0 when one polynomial is divided by another, the divisor and the quotient are factors of the dividend, which means they need to be synthetically divided out again.