Exercise 1672Algebra › Equations and radicalsL1Added 3 Nov 2026

Polynomial divisibility and roots

#polynomial#divisibility#factor theorem

Statement

Find the remainder of the division of $P(x)=x^{100}+x^{50}+1$ by $x^2-1$ .

Hint

$P\equiv P\pmod{x^2-1}$ and the remainder has degree $\le1$ : write $P=(x^2-1)Q+ax+b$ and evaluate at $x=\pm1$ .

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Remainder form. $P=(x^2-1)Q+ax+b$ .

Evaluate at $x=1$ . $P(1)=1+1+1=3=a+b$ .

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

Solve. $a+b=3$ , $-a+b=3$ , so $a=0$ , $b=3$ .

Conclusion. The remainder is $3$ (constant).