Final answer:
The result of the division of x^2 - 3x + 5 by x + 2 using synthetic division is x - 5 with a remainder of 15. The quotient, x - 5, corresponds to choice A) x - 5.
Step-by-step explanation:
The student seems to be asking how to divide the polynomial x^2 - 3x + 5 by x + 2 using synthetic division. To apply synthetic division, we first need to take the root of our divisor, which in this case is x = -2, because we are dividing by x + 2.
We write down the coefficients of the polynomial 1, -3, 5 and bring down the first coefficient (1). We then multiply this coefficient by our root (-2) and write the result below the second coefficient. We continue this process until we've done this for all coefficients. The remainder, if any, is the constant term in the resulting polynomial plus the remainder over our divisor. The synthetic division process is as follows:
- Write the coefficients of x^2 - 3x + 5: 1, -3, 5
- Take the root of the divisor x + 2: -2
- Synthetic division table:
- 1 (bring down)
- 1 * -2 = -2 (multiply and write below the next coefficient)
- -3 - 2 = -5
- -5 * -2 = 10
- 5 + 10 = 15
The result of the synthetic division is the polynomial (1, -5) with a remainder of 15. Therefore, dividing x^2 - 3x + 5 by x + 2 results in x - 5 with a remainder of 15, or x - 5 + 15/(x + 2). The correct answer from the given choices, accounting solely for the quotient and not the remainder, would be A) x - 5.