= Solution
For $P(x)=x^3+4x-7$, the <derivative> $P'(x)=3x^2+4$ is positive everywhere, so the <mean value theorem> makes $P$ strictly increasing. Since a <polynomial> is a <continuous function>, the <intermediate value theorem> and the values $P(1)=-2$, $P(2)=9$ locate its unique real root in $(1,2)$. By the <rational root theorem>, a rational root of a monic <integer> polynomial must be an <integer>. There is no <integer> in $(1,2)$, so \b[the unique real root is an <irrational number>]. The denominator part of that theorem also follows directly: if $a/b$ in lowest terms were a root, $a^3+4ab^2-7b^3=0$ would <force> $b\mid a^3$, hence $b=1$.
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