2025 AMC 10 A

Complete problem set with solutions and individual problem pages

Problem 7 Easy

Suppose a and b are real numbers. When the polynomial x ^ { 3 } + x ^ { 2 } + a x + 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?

  • A.

    14

  • B.

    15

  • C.

    16

  • D.

    17

  • E.

    18

Answer:E

Define f(x) = x^3 + x^2 + ax + b.

Using the given conditions: - f(1) = 1 + 1 + a + b = 4 - f(2) = 8 + 4 + 2a + b = 6

This gives us: - a + b = 2 - 2a + b = -6

Solving: a = -8 and b = 10.

Consequently, b - a = 10 - (-8) = 18.

Related Problems
Problem 5 2025 AMC 10 A
Problem 6 2025 AMC 10 A
Problem 8 2025 AMC 10 A
Problem 9 2025 AMC 10 A
Think Academy Powered by ChatGPT