2025 AMC 12B Problem 8

Problem 8 of 25EasierAlgebra

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?

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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.

Tagged: Vieta’s Formulas · polynomial · radical

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