Prove polynomial identities and use them to solve or describe numerical relationships.
Verify and apply the sum-of-cubes and difference-of-cubes identities
Problem
Verify \(a^3~+~b^3~=~(a~+~b)(a^2~-~ab~+~b^2)\) by expanding the product.
Big Picture
What this problem is really about
A cube identity is easiest to verify by organizing the expansion into two rows instead of juggling six products at once. Distribute each binomial term across the trinomial, align like mixed terms, and watch the sign pattern create two cancellation pairs. If every mixed term disappears, the remaining pure cubes establish the identity.
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