Question

if a+1/a = p and a≠ 0 then show that: a³ + 1/ a³ =p(p² - 3)​

Answer 1

$Given \:a + \frac{1}{a} = p \: ---(1)$

$LHS = a^{3} + \frac{1}{a^{3}} \\= \Big( a + \frac{1}{a}\Big)^{3} -3 \times a \times\frac{1}{a} \Big(a + \frac{1}{a}\Big) \\= \Big( a + \frac{1}{a}\Big)^{3} -3 \times \Big(a + \frac{1}{a}\Big) \\= p^{3} - 3p \:[From \:(1)]$

$= p( p^{2} - 3) \\= RHS$

•••♪