Question

x =3√m+1 +3√m-1/3√m+1-3√m-1 prove that x^3-3mx^2-3x-m

Answer 1

x³ - 3mx² + 3x - m = 0 if x = (∛(m+1)  + ∛(m-1) ) /(∛(m + 1) - ∛(m-1)

Step-by-step explanation:

x = (∛(m+1)  + ∛(m-1) ) /(∛(m + 1) - ∛(m-1)

=> x/1 =  (∛(m+1)  + ∛(m-1) ) /(∛(m + 1) - ∛(m-1))

using componendo & dividendo

=> (x + 1) / (x - 1) = 2∛(m+1) / 2∛(m-1)

=> (x + 1) / (x - 1) = ∛(m+1) / ∛(m-1)

cubing both sides

=> (x + 1)³/(x - 1)³ = (m + 1)/(m -1)

=>  (m + 1)/(m -1) = (x³ + 3x²  + 3x + 1  )/(x³ - 3x² + 3x - 1)

using componendo & dividendo

=> 2m / 2 = 2(x³ + 3x)/2(3x² + 1)

=> m = (x³ + 3x)/(3x² + 1)

=> 3mx² + m = x³ + 3x

=> x³ - 3mx² + 3x - m = 0

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