2025 AMC 10A Problem 7

Question

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

By the Remainder Theorem, evaluate the polynomial at

x = 1

and

x = 2

This gives

a + b + 2 = 4

and

2a + b + 12 = 6

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.

Problem 7 in other years