Binomial theorem & its usage?
In: 2
The binomial theorem tells you how to expand out powers of a binomial, so something like (x+y)^n into a polynomial expression, i.e. a sum of products and powers of x and y. For example, it lets us expand:
– (x+y)^3 = x^3 + 3x^2 •y + 3x•y^2 + y^3
This is important for a number of reasons:
– We know *a lot* about polynomials in two variables. We’ve been studying them for hundreds of years under a variety of different lenses and subjects. We have entire fields dedicated to studying their roots generally, and other fields dedicated to finding integer roots. We can classify their zero-spaces (sets where it’s equal to zero) into families, such as conic sections or elliptic curves, which can be vast oceans of theory in their own right
– The *coefficients* of these expressions (1,3,3,1 in the above example) turn out to be very easy to explicitly calculate. Binomial coefficients are very useful in math, especially in studying a probability distribution where there are only two possible events, like a coin flip or Russian roulette.
– The coefficients are also useful in combinatorics, where the k’th coefficient of some binomial raised to n has a natural interpretation as the number of possible ways to select k of n objects.
Latest Answers