Exercise 1822Algebra › PolynomialsL1Added 3 Nov 2026

GCD of two polynomials

#GCD#Euclidean algorithm#polynomials

Statement

Compute $\gcd(A,B)$ for $A(X)=X^3-1$ and $B(X)=X^2-1$ .

Hint

Apply the Euclidean algorithm.

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First division. $X^3-1=(X^2-1)X+(X-1)$ .

Second division. $X^2-1=(X-1)(X+1)+0$ .

Conclusion. $\gcd(A,B)=X-1$ (up to a nonzero constant).

Check. $X-1$ divides $X^3-1=(X-1)(X^2+X+1)$ and $X^2-1=(X-1)(X+1)$ , and no higher-degree polynomial divides both.