Exercise 333Algebra › PolynomialsL1Added 31 Oct 2026

Remainder of a polynomial

#polynomials#euclidean division

Statement

Find the remainder of the division of $P(x)=x^{100}+x+1$ by $x^2-1$ .

Hint

The remainder has degree at most $1$ . Evaluate $P$ at the roots of $x^2-1$ .

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Write $P(x)=(x^2-1)Q(x)+ax+b$ , where $a,b$ are real numbers.

Evaluating at $x=1$ gives $P(1)=3=a+b$ . Evaluating at $x=-1$ gives $P(-1)=1-1+1=1=-a+b$ .

Adding the two equations gives $2b=4$ , so $b=2$ and $a=1$ . The remainder is $x+2$ .