Binomial (polynomial): Difference between revisions
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*A binomial raised to the ''n<sup>th</sup>'' [[Exponentiation|power]], represented as
::<math> (x + y)^n </math>
:can be expanded by means of the [[binomial theorem]] or, equivalently, using [[Pascal's triangle]]. For example, the square <math>(x+y)^2</math> of the binomial <math>x+y</math> is equal to the sum of the squares of the two terms and twice the product of the terms, that is <math>x^2+2xy+y^2</math>. The numbers (1,2,1) appearing as multipliers for the terms in this expansion are [[binomial coefficient]]s two rows down from the top of Pascal's triangle. The expansion of the ''n''th power uses the numbers ''n'' rows down from the top of the triangle.
*An application of above formula for the square of a binomial is the "(m,n)-formula" for generating [[Pythagorean triple]]s: for ''m < n'', let <math>a=n^2-m^2</math>, <math>b=2mn</math>, <math>c=n^2+m^2</math>, then <math>a^2+b^2=c^2</math>.