AMC 10B Spring 2021 (Problem 15)Expressions & IdentitiesMediumThe real number xxx satisfies the equation x+1x=5x+\frac{1}{x}=\sqrt5x+x1=5. What is the value of x11−7x7+x3x^{11}-7x^7+x^3x11−7x7+x3?(A) −1\text{(A)}\;-1(A)−1(B) 0\text{(B)}\;0(B)0(C) 1\text{(C)}\;1(C)1(D) 2\text{(D)}\;2(D)2(E) 5\text{(E)}\;\sqrt5(E)5Related TopicsCoreAlgebraic IdentitiesCheck AnswerYour answer:ABCDECheckHints (4)Hint 1x11−7x7+x3=x7(x4−7+1x4)x^{11}-7x^7+x^3=x^7\left(x^4-7+\frac{1}{x^4}\right)x11−7x7+x3=x7(x4−7+x41)Hint 2x2+1x2=(x+1x)2−2x⋅1xx^2+\frac{1}{x^2}=\left(x+\frac{1}{x}\right)^2-2x\cdot\frac{1}{x}x2+x21=(x+x1)2−2x⋅x1=(5)2−2=5−2=3=(\sqrt5)^2-2=5-2=3=(5)2−2=5−2=3Hint 3x4+1x4=(x2+1x2)2−2x2⋅1x2x^4+\frac{1}{x^4}=\left(x^2+\frac{1}{x^2}\right)^2-2x^2\cdot\frac{1}{x^2}x4+x41=(x2+x21)2−2x2⋅x21=32−2=7=3^2-2=7=32−2=7Hint 4By Hints 1 and 3,x11−7x7+x3x^{11}-7x^7+x^3x11−7x7+x3=x7(x4−7+1x4)=x^7\left(x^4-7+\frac{1}{x^4}\right)=x7(x4−7+x41)=x7(7−7)=0=x^7(7-7)=0=x7(7−7)=0Final Answer(B) 000Related Problems (6)AMC 10A 2020 (Problem 14)AMC 10B 2025 (Problem 7)AMC 10A 2012 (Problem 22)AMC 10B 2023 (Problem 14)BMO1 2016/2017 (Problem 3)AMC 12B 2025 (Problem 8)