Exercise 537Algebra › PolynomialsHigh schoolAdded 3 Nov 2026

Euclidean division

#Euclidean division#quotient#remainder

Statement

Perform the Euclidean division of $A(X)=X^4-3X^3+2X^2-X+5$ by $B(X)=X^2-2X+1$ .

Hint

Long division: at each step, eliminate the leading term of the current remainder.

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First step. $X^4/(X^2)=X^2$ . Multiply: $X^2(X^2-2X+1)=X^4-2X^3+X^2$ . Subtract from $A$ :
$A-X^2B=-X^3+X^2-X+5.$

Second step. $-X^3/(X^2)=-X$ . Multiply: $-X(X^2-2X+1)=-X^3+2X^2-X$ . Subtract:
$(-X^3+X^2-X+5)-(-X^3+2X^2-X)=-X^2+5.$

Third step. $-X^2/(X^2)=-1$ . Multiply: $-(X^2-2X+1)=-X^2+2X-1$ . Subtract:
$(-X^2+5)-(-X^2+2X-1)=-2X+6.$

Conclusion. Quotient $Q(X)=X^2-X-1$ , remainder $R(X)=-2X+6$ . Check: $A=BQ+R$ .