Exercise 536Algebra › PolynomialsHigh schoolAdded 3 Nov 2026

Divisibility by $X-a$

#factor theorem#divisibility#roots

Statement

Show that $X-1$ divides $P(X)=X^n-1$ for every $n\ge1$ , and find the quotient.

Hint

Use $X^n-1=(X-1)(X^{n-1}+X^{n-2}+\cdots+1)$ .

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Factorization.
$X^n-1=(X-1)(X^{n-1}+X^{n-2}+\cdots+X+1).$

Check. Expanding: the telescoping sum gives $X^n-1$ .

Conclusion. $X-1\mid X^n-1$ , with quotient $X^{n-1}+\cdots+X+1$ .

Remark. This is a special case of the factor theorem: $P(1)=1^n-1=0$ .