Answer:
Sure! Here's the step-by-step process for using synthetic division to divide the polynomial x^2 + 8x + 1 by x - 4:
x - 4 | 1 8 1
--------
1 12 49
Write the dividend in descending order of exponents: x^2 + 8x + 1.
Write the divisor as x - 4.
Set up the synthetic division table with the divisor on the left and the coefficients of the dividend on the top row, including any missing coefficients as zeros:
x - 4 | 1 8 1
Bring down the first coefficient (1) from the top row, and write it in the bottom row under the divisor:
x - 4 | 1 8 1
--------
1
Multiply the divisor (x - 4) by the number in the bottom row (1) to get the result, and write it in the second row under the corresponding coefficients of the dividend:
x - 4 | 1 8 1
--------
1
-4
---
4
Add the result to the next coefficient of the dividend (8) to get the next number in the bottom row:
x - 4 | 1 8 1
--------
1
-4
---
4 <--- this number is the next coefficient
Repeat steps 5 and 6 until all the coefficients have been brought down and the final remainder is obtained. The completed synthetic division table looks like this:
x - 4 | 1 8 1
--------
1
-4
---
4
16
---
49
The bottom row of the table shows the coefficients of the quotient, and the final remainder is written to the right of the division symbol. Therefore, the result of the division is:
x^2 + 8x + 1 = (x - 4)(x + 12) + 49/(x - 4)
So the quotient is x + 12, and the remainder is 49/(x - 4).