Binomial theorem

Algebraic expansion of powers of a binomial

In elementary algebra, the binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. According to the theorem, the power ⁠ ( x + y) n {\displaystyle \textstyle (x+y)^{n}} ⁠ expands into a polynomial with terms of the form ⁠ a x k y m {\displaystyle \textstyle ax^{k}y^{m}} ⁠, where the exponents ⁠ k {\displaystyle k} ⁠ and ⁠ m {\displaystyle m} ⁠ are nonnegative integers satisfying ⁠ k + m = n {\displaystyle k+m=n} ⁠ and the coefficient ⁠ a {\displaystyle a} ⁠ of each term is a specific positive integer depending on ⁠ n {\displaystyle n} ⁠ and ⁠ k {\displaystyle k} ⁠. For example, for ⁠ n = 4 {\displaystyle n=4} ⁠, ( x + y) 4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 x y 3 + y 4 .

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