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

Below is the video solution and professionally curated solution for Problem 15 of the 2021 AMC 10B Spring, 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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Concepts:algebraic manipulation symmetry (algebra)

Difficulty rating: 1340

15.

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

$-1$

$0$

$1$

$2$

$\sqrt{5}$

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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