Advanced Math practice question

Answer: B. −1

Since x − 2 is a factor, p(2) = 0: 8 + 4a + 2b − 12 = 0, so 2a + b = 2. Since division by x + 1 leaves remainder −6, p(−1) = −6: −1 + a − b − 12 = −6, so a − b = 7. Adding 2a + b = 2 and a − b = 7 gives 3a = 9, a = 3, and then b = −4. Thus a + b = −1 (choice B). Choice A (−7) is b − a — the student solves the system correctly but subtracts in the wrong order (or drops a sign when combining the two equations). Choice C (5) comes from evaluating p at x = 1 instead of x = −1 for the divisor x + 1: 1 + a + b − 12 = −6 gives a + b = 5 directly, the classic sign error in the remainder theorem. Choice D (7) reports a − b, the intermediate equation, instead of a + b.

Why the other answers are wrong

A. −7

Your system is right (a = 3, b = −4), but −7 is b − a; the question asks for a + b = 3 + (−4) = −1.

C. 5

You evaluated p at x = 1 instead of x = −1. For the divisor x + 1, the remainder theorem uses the root x = −1, giving −1 + a − b − 12 = −6.

D. 7

7 is the intermediate equation a − b, not the answer; combine it with 2a + b = 2 to get a = 3 and b = −4, so a + b = −1.

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