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If x2-5x+1=0, x3+2x+1x+1x3 = ?

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If x2-5x+1=0, x3+2x+1x+1x3 = ?

by Max@Math Revolution » Mon May 25, 2020 12:25 am

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[GMAT math practice question]

If x^2-5x+1=0, x^3+2(x+1/x)+1/x^3 = ?

A. 90
B. 100
C. 110
D. 120
E. 130
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Re: If x2-5x+1=0, x3+2x+1x+1x3 = ?

by Max@Math Revolution » Wed May 27, 2020 3:10 am
=>

When we divide both sides of the equation x^2 - 5x + 1 = 0 by x, we have x - 5 + 1/x = 0 or x + 1/x = 5.

x^3 + 1/x^3 = (x + 1/x)^3 – 3x(1/x)(x + 1/x) by factoring using sum of cubes.
Then x^3 + 1/x^3 = (x + 1/x)^3 – 3x(1/x)(x + 1/x) = 5^3 – 3·5 = 110.
Thus, x^3 + 2(x + 1/x) +1/x^3 = x^3 + 1/x^3 + 2(x + 1/x) = 110 + 2·5 = 120.

Therefore, D is the answer.
Answer: D
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