Question

A+1/a=p , a is not equal to zero . prove that: a^3+1/a^3= p(p^2-3)

Answer 1

$a + \frac{1}{a} = p \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: ... \: (i)\\ {(a + \frac{1}{a} )}^{3} = {(p)}^{3} \\ {a}^{3} + \frac{1}{ {a}^{3} } + 3(a)( \frac{1}{a} )(a + \frac{1}{a} ) = {p}^{3} \\ {a}^{3} + \frac{1}{ {a}^{3} } + 3(p) = {p}^{3} \: \: \: \: \: \: \: \: \: ... \: (from \: i) \\ {a}^{ 3} + \frac{1}{ {a}^{3} } = {p}^{3} - 3p \\ {a}^{3} + \frac{1}{ {a}^{3} } = p( {p}^{2} - 3)$

... Hence Proved!