Answer: To divide the polynomial (x^4 + 6x^3 - x^2 - 5x + 1) by (x - 2) using synthetic division, follow these steps:
Step 1: Write down the coefficients of the polynomial in descending order of powers of x, including any missing terms with zero coefficients:
1 | 1 6 -1 -5 1
Step 2: Since we are dividing by (x - 2), set x - 2 equal to zero and solve for x:
x - 2 = 0
x = 2
Step 3: Perform the synthetic division using the value we found in Step 2 (x = 2):
2 | 1 6 -1 -5 1
| 2 16 30 50
-----------------
1 8 15 25 51
Step 4: The numbers in the bottom row of the synthetic division represent the coefficients of the quotient polynomial. In this case, the quotient is 1x^3 + 8x^2 + 15x + 25.
The remainder (last number in the bottom row) is 51.
Therefore, the result of the division is:
Quotient: x^3 + 8x^2 + 15x + 25
Remainder: 51