2025 AMC 10A Problem 7

Below is the professionally curated solution for Problem 7 of the 2025 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 10A solutions, or check the answer key.

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Concepts:polynomial system of equations

Difficulty rating: 1250

Suppose $a$ and $b$ are real numbers. When the polynomial $x^3 + x^2 + ax + b$ is divided by $x - 1,$ the remainder is $4.$ When the polynomial is divided by $x - 2,$ the remainder is $6.$ What is $b - a?$

$14$

$15$

$16$

$17$

$18$

Solution:

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

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