Binomial theorem

in falling factorials and differentiate the binomial theorem to prove these identities. Equivalently, alternating binomial coefficients implement a finite difference that annihilates polynomials of degree less than

n

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In a commutative ring of characteristic

p

(x + y)^{p} = x^{p} + y^{p} .

(x + y)^{p^{n}} = x^{p^{n}} + y^{p^{n}}

A

and

B

commute, then

(A + B)^{n} = \sum_{k = 0}^{n} (k n) A^{n - k} B^{k} .

The usual proof works because commutativity allows words with the same numbers of

A

and

B

factors to be collected.

Coefficient extraction from

(1 - x)^{n} (1 + x)^{n} = (1 - x^{2})^{n}

gives

\sum_{r = 0}^{n} (- 1)^{r} (r n)^{2} = {0, (- 1)^{n /2} (n /2 n), n odd, n even .

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The Binomial Theorem is a fundamental result in algebra that provides a formula for expanding expressions of the form $(a + b)^n$, where $n$ is a non-negative integer. The theorem states that: \[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \] In this formula: - $\sum$ denotes summation.