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

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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

Integer coefficients force the conjugate

4 - \sqrt{5}

to be a root as well

The three roots sum to

5,

so find the third root, then expand to read off

a

and

b

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,

r = -3.

Then

(x^2 - 8x + 11)(x + 3)

= x^3 - 5x^2 - 13x + 33,

giving

a = -13

and

b = 33,

a + b = 20.

Thus, the correct answer is C.