The binomial theorem

#binomial theorem#expansion#coefficients

Statement

Expand $(1+x)^4$ using the binomial theorem.

Hint

$(1+x)^n=\sum_{k=0}^{n}\binom{n}{k}x^k$ .

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Formula. $(1+x)^4=\sum_{k=0}^4\binom{4}{k}x^k$ .

Compute. $=\binom{4}{0}+\binom{4}{1}x+\binom{4}{2}x^2+\binom{4}{3}x^3+\binom{4}{4}x^4=1+4x+6x^2+4x^3+x^4$ .

Conclusion. $(1+x)^4=1+4x+6x^2+4x^3+x^4$ .