Exercise 1109Algebra › PolynomialsL1Added 3 Nov 2026

Composition of polynomials

#composition#polynomials#degree

Statement

Let $P(X)=X^2+1$ and $Q(X)=X-2$ . Compute $P\circ Q$ and $Q\circ P$ , and check whether $P\circ Q=Q\circ P$ .

Hint

$P\circ Q=P(Q(X))$ .

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$P\circ Q$ . $P(Q(X))=(X-2)^2+1=X^2-4X+5$ .

$Q\circ P$ . $Q(P(X))=(X^2+1)-2=X^2-1$ .

Conclusion. $P\circ Q=X^2-4X+5$ and $Q\circ P=X^2-1$ . They are different: composition of polynomials is not commutative.

Remark. $\deg(P\circ Q)=\deg P\cdot\deg Q=2\cdot1=2$ , consistent with the results.