Question
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=an+1Cnan−1b+2Cnan−2b2+⋯+n−1Cnabn−1+bn left parenthesis a plus b
Answers
Answered by
GPT-4o mini
The Binomial Theorem states that for any non-negative integer \( n \):
\[
(a + b)^n = \sum_{k = 0}^{n} \binom{n}{k} a^{n-k} b^k
\]
where \( \binom{n}{k} \) is the binomial coefficient, defined as:
\[
\binom{n}{k} = \frac{n!}{k! (n-k)!}
\]
This theorem provides a way to expand the power of a binomial. The formula essentially states that the expansion can be expressed as a sum of terms, where each term consists of a coefficient (the binomial coefficient), followed by powers of \( a \) and \( b \).
To relate your response back to your query, if you see an expansion written as:
\[
(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \ldots + \binom{n}{n} a^0 b^n
\]
This correctly represents the Binomial Theorem. Thus, the established formula for the expansion you mentioned (if fully written out and correctly represents the coefficients and terms involved) follows from and is indeed a representation of the Binomial Theorem.
\[
(a + b)^n = \sum_{k = 0}^{n} \binom{n}{k} a^{n-k} b^k
\]
where \( \binom{n}{k} \) is the binomial coefficient, defined as:
\[
\binom{n}{k} = \frac{n!}{k! (n-k)!}
\]
This theorem provides a way to expand the power of a binomial. The formula essentially states that the expansion can be expressed as a sum of terms, where each term consists of a coefficient (the binomial coefficient), followed by powers of \( a \) and \( b \).
To relate your response back to your query, if you see an expansion written as:
\[
(a+b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + \ldots + \binom{n}{n} a^0 b^n
\]
This correctly represents the Binomial Theorem. Thus, the established formula for the expansion you mentioned (if fully written out and correctly represents the coefficients and terms involved) follows from and is indeed a representation of the Binomial Theorem.