Exercise 1818Algebra › PolynomialsL1Added 3 Nov 2026

Bézout's identity for polynomials

#Bézout#polynomials#linear combination

Statement

Find polynomials $U,V$ with $\deg U<2$ , $\deg V<3$ such that
$U(X)(X^2-1)+V(X)(X^3-1)=X-1.$

Hint

From the previous exercise, $\gcd=X-1$ , so such polynomials exist. Use the Euclidean algorithm's back-substitution.

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From Euclidean algorithm. $X^3-1=(X^2-1)X+(X-1)$ , so
$X-1=(X^3-1)-X(X^2-1).$

Conclusion. $U(X)=-X$ , $V(X)=1$ . Check: $-X(X^2-1)+1\cdot(X^3-1)=-X^3+X+X^3-1=X-1$ . Correct.

Remark. The degrees are $\deg U=1<2$ and $\deg V=0<3$ , as required.