Final answer:
The division of the polynomial (-2x³ + 5x²- x + 2) by (x + 2) using synthetic division results in a quotient of -2x² + 9x - 19 with a remainder of 40. B: -2x² + 9x - 19, R 40
Step-by-step explanation:
When dividing the polynomial (-2x³ + 5x²- x + 2) by (x + 2) using synthetic division, the first step is to identify the 'zero' of the divisor. For x + 2, this zero is -2 as x = -2 makes the divisor zero. We list the coefficients of the polynomial which are -2, 5, -1, 2 for each power of x in descending order. Then we begin the synthetic division process:
- Bring down the leading coefficient (-2).
- Multiply this by the zero (-2) and put the result under the next coefficient (5).
- Add the column (5 + 4 = 9)
- Repeat this multiplication and addition process for the entire row.
The results give the coefficients of the quotient polynomial. Our final answer will be a polynomial of one degree less than the original polynomial. In this case, since we start with a cubic polynomial, our quotient will be a quadratic polynomial.
After completing the division, we're left with a quotient of -2x² + 9x - 19 and a remainder, which is determined by repeating the process with the final column. If done correctly, the remainder should be +40 or -36. We can discount options A and D which have incorrect quotients or remainders.
The correct answer is option B: -2x² + 9x - 19, R 40.