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Did you know that the expression \( (x+2)^{3} \) can be expanded using the Binomial Theorem? This theorem reveals that \( (a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \). For our expression, this translates to \( x^3 + 3(2)x^2 + 3(2^2)x + 2^3 \), which simplifies to \( x^3 + 6x^2 + 12x + 8 \). It’s a fantastic way to see the power of algebra! If you ever find yourself needing to tackle polynomials like this, remember to watch out for common mistakes! A frequent pitfall is forgetting to correctly apply the powers to each term when expanding. For instance, it’s easy to mistakenly compute \( 2^3 \) as \( 6 \) instead of \( 8 \). Always double-check your calculations while keeping track of all the terms to avoid losing points on homework or exams!
