Question

If a+1/a = p, probe that a³+1/a³ = p (p²-3).​

Answer 1

$\bigstar\rm\blue{GIVEN}\bigstar$

• If $a+\dfrac{1}{a}=p$......1

$\bigstar\sf\blue{TO\:PROVE}\bigstar$

• a³+1/a³ = p (p²-3).

Now,

Cubing the both sides of eq 1

$\implies\sf{(a+\dfrac{1}{a})^{3}=p^{3}}$

$\implies\sf{a+\dfrac{1}{a}^{3}+3\times{{\cancel{a}}}\times{\dfrac{1}{\cancel{a}}}(a+\frac{1}{a})}$

$\implies\sf{(a+\dfrac{1}{a}^{3})+3p=p^3}$

$\implies\sf{a+\dfrac{1}{a}^{3}=p^3-3p}$

$\implies\sf{a+\dfrac{1}{a}^{3}=p(p^2-3)}$

Hence, Proved.