Question

Use synthetic division to find the quotient and remainder when -x^4 + 2x^3 + 12x^2 - 16 is divided by x + 2.

a. Quotient: -x^3 + 2x^2 + 16x - 48, Remainder: 96
b. Quotient: -x^3 + 2x^2 + 16x - 8, Remainder: -8
c. Quotient: -x^3 + 2x^2 + 12x - 8, Remainder: -16
d. Quotient: -x^3 + 2x^2 + 12x - 16, Remainder: 0

Answer 1

When dividing the polynomial -x^4 + 2x^3 + 12x^2 - 16 by x + 2 using synthetic division, we find the quotient to be -x^3 + 2x^2 + 12x - 8 and the remainder to be -16.

Step-by-step explanation:

We are asked to use synthetic division to find the quotient and remainder of the division of the polynomial -x^4 + 2x^3 + 12x^2 - 16 by x + 2. Let's perform the synthetic division step-by-step:

1. First, we write down the coefficients of the polynomial: -1, 2, 12, 0 (since there is no x term), and -16.
2. Next, we write the zero of the divisor x + 2, which is -2.
3. We bring down the first coefficient (-1).
4. We multiply -2 by -1 and write the result (2) under the second coefficient (2).
5. We then add the second coefficient and the result (2 + 2 = 4) and write it below.
6. This process is repeated for all coefficients: multiply with -2, write the result, add it to the next coefficient.
7. Finally, we get the coefficients of the quotient and the remainder.

Following these steps, we find the quotient to be -x^3 + 2x^2 + 12x - 8, and the remainder is -16. Therefore, the correct answer is option c.