algebra

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If \(x - \frac{1}{x} = 5\) then find \(x^3 - \frac{1}{x^3}\)

Guest Jul 28, 2020

geno3141 · Answer 1 · 2020-07-28T10:39:26

Since x - 1/x = 5 ---> (x - 1/x)³ = 5³

Multiplying out:

---> x³ - 3(x)²(1/x) + 3(x)(1/x)² - 1/x³ = 125

---> [x³ - 1/x³] + [ - 3(x)²(1/x) + 3(x)(1/x)² ] = 125

But: -3(x)²(1/x) + 3(x)(1/x)² = -3x + 3/x = -3(x - 1/x) = -3(5) = -15

So: [x³ - 1/x³] + [ - 3(x)²(1/x) + 3(x)(1/x)² ] = 125

---> [x³ - 1/x³] + [ - 15 ] = 125

---> x³ - 1/x³ = 140