Let $a$ and $b$ be two real numbers such that $a+b=7$ and $a b=3$. What is the value of $a^{3}+b^{3}$? We will not seek to express $a$ and $b$.

Let's factorize: $a^{3}+b^{3}=(a+b)\left(a^{2}-a b+b^{2}\right)$.

It remains to express $\left(a^{2}-a b+b^{2}\right)$ in terms of $a+b$ and $a b$. To make the $a^{2}$ and $b^{2}$ appear, we can use $(a+b)^{2}$. We have $a^{2}-a b+b^{2}=a^{2}+2 a b+b^{2}-3 a b=(a+b)^{2}-3 a b$.

Therefore, $a^{3}+b^{3}=(a+b)\left((a+b)^{2}-3 a b\right)=7\left(7^{2}-9\right)=7 \times 40=280$.