In elementary algebra, the binomial theorem or the binomial expansion is a mechanism by which expressions of the form (x + y)n can be expanded. It is the identity that states that for any non-negative integer n,
is a binomial coefficient. Another useful way of stating it is the following:
An alternate way to find the binomial coefficients is by using Pascal's triange. The triangle is built from apex down, starting with the number one alone on a row. Each number is equal to the sum of the two numbers directly above it.
n=0 1 n=1 1 1 n=2 1 2 1 n=3 1 3 3 1 n=4 1 4 6 4 1 n=5 1 5 10 10 5 1
Thus, the binomial coefficients for the expression (x + y)4 are 1, 3, 6, 4, and 1.
One way to prove this identity is by mathematical induction.
Base case: n = 0
Induction case: Now suppose that it is true for n : and prove it for n + 1.
and the proof is complete.
These are the expansions from 0 to 6.
Newton's binomial theorem
There is also Newton's binomial theorem, proved by Isaac Newton, that goes beyond elementary algebra into mathematical analysis, which expands the same sum (x + y)n as an infinite series when n is not an integer or is not positive.
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