Binomial Theorem

The most basic example of the binomial theorem is the formula for the square of x + y:

${\displaystyle (x+y)^{2}=x^{2}+2xy+y^{2}.}$

The binomial coefficients 1, 2, 1 appearing in this expansion correspond to the second row of Pascal's triangle. (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:

${\displaystyle {\begin{aligned}(x+y)^{3}&=x^{3}+3x^{2}y+3xy^{2}+y^{3},\\[8pt](x+y)^{4}&=x^{4}+4x^{3}y+6x^{2}y^{2}+4xy^{3}+y^{4},\\[8pt](x+y)^{5}&=x^{5}+5x^{4}y+10x^{3}y^{2}+10x^{2}y^{3}+5xy^{4}+y^{5},\\[8pt](x+y)^{6}&=x^{6}+6x^{5}y+15x^{4}y^{2}+20x^{3}y^{3}+15x^{2}y^{4}+6xy^{5}+y^{6},\\[8pt](x+y)^{7}&=x^{7}+7x^{6}y+21x^{5}y^{2}+35x^{4}y^{3}+35x^{3}y^{4}+21x^{2}y^{5}+7xy^{6}+y^{7}.\end{aligned}}}$

Several patterns can be observed from these examples. In general, for the expansion (x + y)ⁿ:

1. the powers of x start at n and decrease by 1 in each term until they reach 0 (with x⁰ = 1, often unwritten);
2. the powers of y start at 0 and increase by 1 until they reach n;
3. the nth row of Pascal's Triangle will be the coefficients of the expanded binomial when the terms are arranged in this way;
4. the number of terms in the expansion before like terms are combined is the sum of the coefficients and is equal to 2ⁿ; and
5. there will be n + 1 terms in the expression after combining like terms in the expansion.

The binomial theorem can be applied to the powers of any binomial. For example,

${\displaystyle {\begin{aligned}(x+2)^{3}&=x^{3}+3x^{2}(2)+3x(2)^{2}+2^{3}\\&=x^{3}+6x^{2}+12x+8.\end{aligned}}}$

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:

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