examday

/TMUA

Let $a$ and $b$ be nonzero real numbers such that $a+b$ and $a^3+b^3$ are rational. A student gives the following proof that $ab$ is rational.

1. Write $s=a+b$ and $t=a^3+b^3$. Then $s$ and $t$ are rational.
2. Expanding $s^3$ gives $s^3=t+3abs$.
3. Since $a\ne0$ and $b\ne0$, we have $s\ne0$.
4. Hence $ab=(s^3-t)/(3s)$ is rational.

Which of the following is correct?