There are integers a and b such that the polynomial x^3 - 5x^2 + ax + b has 4 + √5 as a root. What is a + b?

2025 AMC 12B Problem 8 — Solution

2025 AMC 12B Problem 8

Attempt Problem 8 of the 2025 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2025 AMC 12B solutions, or check the answer key.

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There are integers $a$ and $b$ such that the polynomial $x^3 - 5x^2 + ax + b$ has $4 + \sqrt{5}$ as a root. What is $a + b?$

$13$

$17$

$20$

$30$

$68$

Answer: C

Concepts:Vieta’s Formulas polynomial radical

Difficulty rating: 1440

Solution:

The conjugate $4 - \sqrt{5}$ is also a root, and these two are the roots of $x^2 - 8x + 11.$ The third root $r$ satisfies $8 + r = 5,$ so $r = -3.$ Then $(x^2 - 8x + 11)(x + 3)$ $= x^3 - 5x^2 - 13x + 33,$ giving $a = -13$ and $b = 33,$ so $a + b = 20.$

Thus, the correct answer is C.

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