Question

If $p+q=5$ and $pq=6$, then $p^3+q^3=$

Answer 1

Using the identity $(a+b{)}^3=a^3+b^3+3ab(a+b)$, we can write

${(p+q)}^3=p^3+q^3+3pq(p+q)$

It is given that $p+q=5$ and $pq=6$, therefore,

${(p+q)}^3=p^3+q^3+3pq(p+q)\Rightarrow {\left(5\right)}^3=p^3+q^3+(3\times 6)\left(5\right)\Rightarrow 125=p^3+q^3+(18\times 5)\Rightarrow 125=p^3+q^3+90\Rightarrow p^3+q^3=125-90\Rightarrow p^3+q^3=35$

Hence, $p^3+q^3=35$.