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
( the x-2 in my answer is meant to go under the remainder )
To divide a polynomial synthetically, follow these steps:
1. Set up the division problem by writing the dividend (the polynomial being divided) and the divisor. In this case:
Dividend: x^4 - x^3 + x^2 - x + 2
Divisor: x - 2
2. Begin the synthetic division by focusing on the coefficients of the terms in the dividend. Write them in descending order, including any zero coefficients:
Coefficients: 1, -1, 1, -1, 2
3. Start with the first coefficient, which is 1. Bring it down below the divisor:
x - 2 | 1 - 1 1 - 1 2
4. Multiply the divisor (x - 2) by the first term (1). Write the result below the next coefficient:
x - 2 | 1 - 1 1 - 1 2
| 2
5. Add the numbers in the second row, which represents the multiplication result, to the coefficients in the first row:
x - 2 | 1 - 1 1 - 1 2
| 2
-------------
1 1 0 - 2 4
6. Repeat steps 4 and 5 until you reach the last coefficient. In this case, you need to repeat twice more.
7. Continue the process:
x - 2 | 1 - 1 1 - 1 2
| 2 - 2
-------------
1 1 0 - 2 4
-----------
4 - 4
8. The last row of numbers represents the coefficients of the quotient and the remainder in decreasing order. From left to right, these coefficients are:
Quotient: 1, 1, 0
Remainder: -2, 4
Therefore, the correct answer to the division problem is:
Quotient: x^2 + x
Remainder: -2x + 4
Please note that your answer, x^3 + x^2 + 3x + 5 + 12 (remainder), appears to have an incorrect quotient.