Exercise 1669Algebra › Equations and radicalsL1Added 3 Nov 2026

A functional equation

#functional equation#polynomial solutions#algebraic

Statement

Find all polynomial functions $P$ with real coefficients such that
$P(x+1)-P(x)=2x+1$
for all $x\in\mathbb{R}$ .

Hint

Compare degrees: the LHS has degree $\deg P-1$ and the RHS has degree $1$ .

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Degree. If $\deg P=n$ , then $\deg(P(x+1)-P(x))=n-1$ (leading terms cancel). So $n-1=1$ , $n=2$ .

Form. Write $P(x)=ax^2+bx+c$ . Then
$P(x+1)-P(x)=a(x+1)^2+b(x+1)+c-ax^2-bx-c=2ax+a+b.$

Identify. $2a=2$ , so $a=1$ . $a+b=1$ , so $b=0$ . $c$ is arbitrary.

Conclusion. $P(x)=x^2+c$ for any $c\in\mathbb{R}$ .

Check. $P(x+1)-P(x)=(x+1)^2+c-(x^2+c)=2x+1$ . Correct.