2021 AMC 10B Problem 15

Problem 15 of 25IntermediateAlgebra

The real number

x

satisfies the equation

x+\frac{1}{x} = \sqrt{5}.

What is the value of

x^{11}-7x^{7}+x^3?

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Solution

Since

x+\frac{1}{x} = \sqrt{5},

squaring yields

x^2 + 2 + \frac 1{x^2} = 5

x^2 + \frac 1{x^2} = 3.

Squaring again yields

x^4 + 2 + \frac 1{x^4} = 9

x^4 -7 + \frac 1{x^4} = 0.

Multiplying by

x^7

yields

x^{11}-7x^{7}+x^3=0.

Thus, the answer is B .

Concepts: algebraic manipulation · symmetry (algebra)

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.