To find the quotient using synthetic division, we first need to clarify the expression. It appears there might be a slight misunderstanding in the provided polynomial. Assuming you meant the polynomial to be 3x3 + 32x2 + 4x, we can proceed with synthetic division by a factor of the form (x – c).
Let’s choose a value for c. For example, if we are dividing by x – 2, we would set up the synthetic division as follows:
1. Write down the coefficients of the polynomial: 3, 32, 4, 0 (note the zero for the missing term).
2. Write the root 2 on the left.
2. Bring down the leading coefficient (3) directly below the line.
3. Multiply the root (2) by the value you just brought down (3) and write the result (6) underneath the next coefficient (32).
4. Add the numbers in the second column: 32 + 6 = 38.
5. Repeat the process: multiply 2 by 38, resulting in 76, and write it under the next coefficient (4).
6. Finally, add these numbers: 4 + 76 = 80.
At this point, your synthetic division should look like this:
2 | 3 32 4 0
| 6 76
----------------
3 38 80
The bottom row (3, 38) gives us the coefficients of the quotient, which corresponds to 3x^2 + 38x and the remainder (80).
Thus, the result of the synthetic division yields a quotient of 3x2 + 38x with a remainder of 80 when divided by (x – 2).