I need to divide this polynomial synthetically. I think I did it right, but can someone confirm if my answer is right or wrong?
Polynomial:
x^4 - x^3 + x^2 - x + 2
Divisor: x-2
My answer:
x^3 + x^2 + 3x + 5 + 12 (remainder)
----
x-2
To confirm if your answer is right or wrong, we can perform a synthetic division using the given polynomial and divisor. Here are the steps:
Step 1: Write the polynomial in descending order of powers, including zero placeholders for missing terms:
x^4 - x^3 + x^2 - x + 2
Step 2: Set up the synthetic division as follows, with the divisor on the left side and the polynomial's coefficients on the top:
2 | 1 -1 1 -1 2
Step 3: Bring down the first coefficient (1) directly below the line:
2 | 1 -1 1 -1 2
-----------------
1
Step 4: Multiply the divisor (2) by the number brought down (1) and write the result below the next coefficient (1):
2 | 1 -1 1 -1 2
-----------------
1
------
2
Step 5: Add the result (2) to the next coefficient (-1) and write the sum below:
2 | 1 -1 1 -1 2
-----------------
1
------
2
2
Step 6: Repeat steps 4 and 5 until all coefficients are used:
2 | 1 -1 1 -1 2
-----------------
1
------
2
2 -2
----------
1 1
Step 7: The final line represents the coefficients of the quotient:
Quotient: 1 1 2
Step 8: The remainder is the last number in the final line:
Remainder: 1
Therefore, your answer is incorrect. The correct quotient is x^3 + x^2 + 2, and the remainder is 1.