Exercise 1169Algebra › Rings and fieldsL1Added 3 Nov 2026

Euclidean division in $\mathbb{R}[X]$

#Euclidean division#polynomials#quotient and remainder

Statement

In $\mathbb{R}[X]$ , perform the Euclidean division of $A=X^3-2X^2+X-3$ by $B=X-2$ .

Hint

Use the factor theorem: the remainder of division by $X-a$ is $A(a)$ .

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Remainder. $A(2)=8-8+2-3=-1$ . So the remainder is $-1$ .

Quotient. $A=(X-2)Q-1$ . To find $Q$ , perform the division:
$X^3-2X^2+X-3=(X-2)(X^2+1)-1.$
So $Q=X^2+1$ .

Check. $(X-2)(X^2+1)-1=X^3+X-2X^2-2-1=X^3-2X^2+X-3$ . Correct.

Conclusion. $A=(X-2)(X^2+1)-1$ , quotient $X^2+1$ , remainder $-1$ .