Expanding with the Binomial Theorem
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Expanding with the Binomial Theorem means writing the coefficient row, walking one base's exponent down while the other's climbs, and simplifying each term.
Expand (x + 2)³. The row for n = 3: 1, 3, 3, 1. The powers of x fall 3, 2, 1, 0 while the powers of 2 rise 0, 1, 2, 3. Assemble: x³ + 3x²(2) + 3x(4) + 8, which simplifies to x³ + 6x² + 12x + 8. The audit is one substitution: at x = 1, the original is 3³ = 27, and the expansion answers 1 + 6 + 12 + 8 = 27. Exponents in every term sum to n — a second free check — and the term count is n + 1, never n. Any binomial expands this way, subtraction included: a − b runs the same row with alternating signs.
Raise the whole second base to its power; the 2³ is 8, not 6.
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