Prove polynomial identities and use them to solve or describe numerical relationships.
Decide whether a polynomial equation is an identity by expanding both sides and comparing
Problem
Is this equation an identity? \((x~+~1)^2~=~x^2~+~2x~+~1\)
Big Picture
What this problem is really about
An identity claims equality for every allowed input, so appearance alone cannot settle it. Expand and simplify each side until both are written in standard polynomial form, then compare corresponding coefficients. Matching polynomials prove the claim universally; by contrast, one failed test value can disprove an identity, while one successful test value cannot prove it.
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