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2025 AMC 10A Problem 7

Problem 7 of 25EasierAlgebra

Suppose aa and bb are real numbers. When the polynomial x3+x2+ax+bx^3 + x^2 + ax + b is divided by x−1,x - 1, the remainder is 4.4. When the polynomial is divided by x−2,x - 2, the remainder is 6.6. What is b−a?b - a?

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Solution

By the Remainder Theorem, just plug in. We get p(1)=1+1+a+b=4,p(1) = 1 + 1 + a + b = 4, so a+b=2.a + b = 2. And p(2)=8+4+2a+b=6,p(2) = 8 + 4 + 2a + b = 6, so 2a+b=−6.2a + b = -6. Subtract the first from the second: a=−8,a = -8, hence b=10.b = 10. Then b−a=10−(−8)=18.b - a = 10 - (-8) = 18. Thus, E is the correct answer.

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Concepts: polynomial · system of equations

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.