Which of the following establishes the Binomial Theorem for the expansion of polynomials in the form (a+b)n ?(1 point) Responses (a+b)n=bn+nC1abn−1+nC2a2bn−2+⋯+nCn−1an−1b+an left parenthesis a plus b right parenthesis superscript n baseline equals b superscript n baseline plus subscript n baseline upper C subscript 1 baseline a b superscript n minus 1 baseline plus subscript n baseline upper C subscript 2 baseline a squared b superscript n minus 2 baseline plus dot dot dot plus subscript n baseline upper C subscript n minus 1 baseline a superscript n minus 1 baseline b plus a superscript n baseline (a+b)n=an+nC1an−1b+nC2an−2b2+⋯+nCn−1abn−1+bn left parenthesis a plus b right parenthesis superscript n baseline equals a superscript n baseline plus subscript n baseline upper C subscript 1 baseline a superscript n minus 1 baseline b plus subscript n baseline upper C subscript 2 baseline a superscript n minus 2 baseline b squared plus dot dot dot plus subscript n baseline upper C subscript n minus 1 baseline a b superscript n minus 1 baseline plus b superscript n baseline (a+b)n=an+1Cnan−1b+2Cnan−2b2+⋯+n−1Cnabn−1+bn left parenthesis a plus b right parenthesis superscript n baseline equals a superscript n baseline plus subscript 1 baseline upper C subscript n baseline a superscript n minus 1 baseline b plus subscript 2 baseline upper C subscript n baseline a superscript n minus 2 baseline b squared plus dot dot dot plus subscript n minus 1 baseline upper C subscript n baseline a b superscript n minus 1 baseline plus b superscript n baseline (a+b)n=an+bn

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The correct form that establishes the Binomial Theorem for the expansion of polynomials in the form \((a+b)^n\) is:

\[ (a+b)^n = a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a b^{n-1} + b^n \]

This means the correct response is:

\((a+b)^n = a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a b^{n-1} + b^n\)

This shows the complete expansion of the binomial expression using the coefficients \(\binom{n}{k}\) (which are known as binomial coefficients) for \(k = 0\) to \(n\).

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Which of the following establishes the Binomial Theorem for the expansion of polynomials in the form (a+b)n ?(1 point) Responses (a+b)n=bn+nC1abn−1+nC2a2bn−2+⋯+nCn−1an−1b+an left parenthesis a plus b right parenthesis superscript n baseline equals b superscript n baseline plus subscript n baseline upper C subscript 1 baseline a b superscript n minus 1 baseline plus subscript n baseline upper C subscript 2 baseline a squared b superscript n minus 2 baseline plus dot dot dot plus subscript n baseline upper C subscript n minus 1 baseline a superscript n minus 1 baseline b plus a superscript n baseline (a+b)n=an+nC1an−1b+nC2an−2b2+⋯+nCn−1abn−1+bn left parenthesis a plus b right parenthesis superscript n baseline equals a superscript n baseline plus subscript n baseline upper C subscript 1 baseline a superscript n minus 1 baseline b plus subscript n baseline upper C subscript 2 baseline a superscript n minus 2 baseline b squared plus dot dot dot plus subscript n baseline upper C subscript n minus 1 baseline a b superscript n minus 1 baseline plus b superscript n baseline (a+b)n=an+1Cnan−1b+2Cnan−2b2+⋯+n−1Cnabn−1+bn left parenthesis a plus b right parenthesis superscript n baseline equals a superscript n baseline plus subscript 1 baseline upper C subscript n baseline a superscript n minus 1 baseline b plus subscript 2 baseline upper C subscript n baseline a superscript n minus 2 baseline b squared plus dot dot dot plus subscript n minus 1 baseline upper C subscript n baseline a b superscript n minus 1 baseline plus b superscript n baseline (a+b)n=an+bn

1 answer

The correct form that establishes the Binomial Theorem for the expansion of polynomials in the form \((a+b)^n\) is:

\[ (a+b)^n = a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a b^{n-1} + b^n \]

This means the correct response is:

\((a+b)^n = a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a b^{n-1} + b^n\)

This shows the complete expansion of the binomial expression using the coefficients \(\binom{n}{k}\) (which are known as binomial coefficients) for \(k = 0\) to \(n\).

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GPT-4o mini · Answer

The correct form that establishes the Binomial Theorem for the expansion of polynomials in the form \((a+b)^n\) is:

\[ (a+b)^n = a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a b^{n-1} + b^n \]

This means the correct response is:

\((a+b)^n = a^n + \binom{n}{1} a^{n-1} b + \binom{n}{2} a^{n-2} b^2 + \cdots + \binom{n}{n-1} a b^{n-1} + b^n\)

This shows the complete expansion of the binomial expression using the coefficients \(\binom{n}{k}\) (which are known as binomial coefficients) for \(k = 0\) to \(n\).