Newton binomial formula

a binomial. According to the theorem, it is possible to expand

the power (x + y)n into a sum involving terms of the form a xb yc, where the exponents band c are nonnegative integers with b + c = n, and the coefficient a of each term is a specific positive integer depending on n and b. For example,

work of 11th-century Persian mathematician Al-Karaji who described the triangular pattern of

the binomial coefficients. He also provided a mathematical proof of both the binomial theorem and Pascal's triangle, using a primitive form of mathematical induction. The Persian poet and mathematician Omar Khayyam was probably familiar with the formula to higher orders, although many of his mathematical works are lost..The binomial expansions of small degrees were known in the 13th century mathematical works of Yang Hui and also Chu Shih-Chieh. Yang Hui attributes the method to a much earlier 11th century text of Jia Xian, although those writings are now also lost.

power of x + y into a sum of the form
where each   is a

specific positive integer known as a binomial coefficient. (When an exponent is zero, the corresponding power expression is taken to be 1 and this multiplicative factor is often omitted from the term. Hence one often sees the right side written as  This formula is also referred to as the binomial formula or the binomial identity. Using summation notation, it can be written as

symmetry of x and y in the first expression, and

by comparison it follows that the sequence of binomial coefficients in the formula is symmetrical. A simple variant of the binomial formula is obtained by substituting 1 for y, so that it involves only a single variable. In this form, the formula reads

or equivalently

formula for the  square  of x + y:
The binomial coefficients 1, 2, 1

appearing in this expansion correspond to the second row of Pascal's triangle. (Note that the top "1" of the triangle is considered to be row 0, by convention.) The coefficients of higher powers of x + y correspond to lower rows of the triangle:

for the expansion (x + y)n:
the powers of x start at n and decrease by 1 in

each term until they reach 0 (with {{{1}}} often unwritten);
the powers of y start at 0 and increase by 1 until they reach n;
the nth row of Pascal's Triangle will be the coefficients of the expanded binomial when the terms are arranged in this way;
the number of terms in the expansion before like terms are combined is the sum of the coefficients and is equal to 2n; and
there will be n + 1 terms in the expression after combining like terms in the expansion.

any binomial. For example,
For a binomial involving subtraction, the theorem

can be applied by using the form (x − y)n = (x + (−y))n. This has the effect of changing the sign of every other term in the expansion: