AMC 10 · 2025 · #7

Grade 8 algebra
Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search
polynomial-remainderfunction-evaluationsystems-of-equations convert-to-algebra ↑ Prerequisites: polynomial-remainder
📏 Medium solution 💡 2 insights
Problem
The cubic f(x) = x³ + x² + ax + b leaves remainder 4 when divided by x - 1 and remainder 6 when divided by x - 2. Find b - a.

Pick an answer.

(A)
14
(B)
15
(C)
16
(D)
17
(E)
18

AMC 10 2025 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

Long division looks like the obvious route, but the Remainder Theorem lets you skip it entirely: dividing f(x) by x - c leaves remainder f(c), so each remainder fact is just the value of f at a point. Tool #7 (Identify Subproblems) sees the two divisions as two independent clues, and Tool #4 (Introduce a Variable) turns each clue into an equation in the unknowns a and b. Two equations, two unknowns — solve the pair and read off b - a.

1STEP 1

Read remainders as values

By the Remainder Theorem, dividing by x - c leaves f(c), so the two facts become f(1) = 4 and f(2) = 6 — no long division.

f(x) = (x - c) q(x) + r → f(c) = r, f(1) = 4, f(2) = 6
2STEP 2

Evaluate at x = 1

Substituting x = 1 gives 1 + 1 + a + b = 4, which collapses to the linear equation a + b = 2.

f(1) = 1 + 1 + a + b = 4 → a + b = 2 i
3STEP 3

Evaluate at x = 2

Substituting x = 2 gives 8 + 4 + 2a + b = 6, so the second equation is 2a + b = -6.

f(2) = 8 + 4 + 2a + b = 6 → 2a + b = -6 ii
4STEP 4

Solve the two equations

Subtracting the first equation from the second cancels b and isolates a = -8; putting that back gives b = 10.

(2a + b) - (a + b) = -6 - 2 → a = -8, -8 + b = 2 → b = 10
5STEP 5

Compute b − a

The ask is b - a, not a or b alone; subtracting a negative adds, so 10 - (-8) = 18, choice (E).

b - a = 10 - (-8) = 18 (E)
Answer
18
Rebuild the polynomial with a = -8 and b = 10: f(x) = x³ + x² - 8x + 10. Then f(1) = 1 + 1 - 8 + 10 = 4 and f(2) = 8 + 4 - 16 + 10 = 6 — both remainders come out exactly as stated, so the values are correct and b - a = 18.
💡Key takeaway

A remainder after dividing by x - c is just the value f(c), so plug in each number to build equations instead of doing long division.

  • Read remainders as values
  • Evaluate at x = 1
  • Evaluate at x = 2
  • Solve the two equations
  • Compute b − a

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