rational function question

if x=a－1/a，express a^3－1/a^3，in terms of x
Sol
a^3－1/a^3
=(a－1/a)(a^2+1+1/a^2)
=(a－1/a)(a^2－2+1/a^2+3)
=x(x^2+3)

x=a-1/a
x³=a³-3a+3/a-1/a³
a³-1/a³=x³+3a-3/a=x³+3x

x = a - (1/a)^2
x^2
= [a - (1/a)]^2
= a^2 - 2 + (1/a)^2
= a^2 + (1/a)^2 - 2

so
a^3 - (1/a^3)
= a^3 - (1/a)^3

[by formula a^3 - b^3 = (a - b)(a^2 + ab + b^2)
= [a - (1/a)][a^2 + 1 + (1/a)^2]
= x [a^2 + (1/a)^2 - 2 + 3]
= x (x^2 + 3)

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x = a - 1/a
x^2
= (a - 1/a)^2
= a^2 - 2a(1/a) + 1/a^2
= a^2 + 1/a^2 - 2
a^2 + 1/a^2 = x^2 + 2

a^3 - 1/a^3
= (a - 1/a)(a^2 + (a)(1/a) + 1/a^2)
= (a - 1/a)(a^2 + 1/a^2 + 1)
= x(x^2 + 2 + 1)
= x(x^2 + 3)