2021 AMC 10B Problem 15

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 .