Exercise 84Number theory › IrrationalityL1Added 3 Nov 2026

Rational root theorem and irrationality

#rational root theorem#irrationality#cubic

Statement

Show that $X^3-X-1$ has no rational roots, hence any real root is irrational.

Hint

Rational roots must be $\pm1$ . Test both.

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Candidates. $P(X)=X^3-X-1$ is monic with constant term $-1$ , so rational roots are $\pm1$ .

Test. $P(1)=1-1-1=-1\neq0$ . $P(-1)=-1+1-1=-1\neq0$ .

Conclusion. No rational root. Since the polynomial has real roots (odd degree), they are irrational.

Approximation. The real root is $\approx1.3247$ (plastic constant).