Final answer:
Using synthetic division, the polynomial 2x^3 - x^2 - 12 divided by x + 3 results in 2x^2 - 7x + 21 with a remainder of -51, which corresponds to option (b).
Step-by-step explanation:
The student has asked to use synthetic division to divide the polynomial 2x^3 - x^2 - 12 by x + 3. To verify which of the options (a) 2x^2 - 7x + 18, (b) 2x^2 - 7x - 21, (c) 2x^2 - 5x + 18, or (d) 2x^2 - 5x - 15 is correct, we perform synthetic division.
To start, we write down the coefficients of the polynomial: 2, -1, 0, -12 (note the 0 is for the x term which is missing). The synthetic division process will look like this:
Write the root of the divisor x + 3, which is -3, on the left.
Beneath the line write the coefficients: 2, -1, 0, -12.
Bring down the 2 to the bottom row.
Multiply -3 by 2 and write the result under the next coefficient -1. The result is -6. Add -1 and -6 to get -7 and write this number under the line.
Multiply -3 by -7 to get 21, write it under the 0, add to get 21.
Finally, multiply -3 by 21 to get -63, and add it to -12 to get -51, which is the remainder.
The result of synthetic division is 2x^2 - 7x + 21 with remainder -51. This corresponds with option b).