The real number x satisfies the equation x+(1)/(x) = √5. What is the value of x^11-7x^7+x^3?

2021 AMC 10B Spring Problem 15 — Solution & Video

2021 AMC 10B Spring Problem 15

Attempt Problem 15 of the 2021 AMC 10B Spring below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 10B Spring solutions, or check the answer key.

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15.

The real number $x$ satisfies the equation $x+\frac{1}{x} = \sqrt{5}.$ What is the value of $x^{11}-7x^{7}+x^3?$

$-1$

$0$

$1$

$2$

$\sqrt{5}$

Answer: B

Concepts:algebraic manipulation symmetry (algebra)

Difficulty rating: 1340

Video solution:

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Written 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 .

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