Binomials


 * __ Introduction to Binomials__**

You are now probably wondering, what is a binomial? A binimial is an expression which has two unlike terms. For example x+y is a binomial and so is 3u+k or p(2+12t).

The expansion of (x+y)^n is called __binomial expansion__. The coefficients obtained in the expanded equation are called __binomial coefficients__.

There are the __binomial expansions__ of 2, 3, 4 and 5.



__There are a few key points to note about this:__ When the power of a binomial is n, the number of terms in the expression is (n+1) For every term in the expression, the sum of the powers of x and y is n. When the expression is expanded in the above fashion, the power of x decreases while the power of y increases as we move along the terms.

The coefficients can form a pattern as shown below. This is known as the __Pascal's Triangle__.

(x+y)^0 - 1 (x+y)^1 - 1 - 1 (x+y)^2 - 1 - 2 - 1 (x+y)^3 - 1 - 3 - 3 - 1 (x+y)^4 - 1 - 4 - 6 - 4 - 1 (x+y)^5 - 1 - 5 10 10 5 1 (x+y)^6 - 1 - 6 15 20 15 6 1

Every number is the result of adding the two numbers that are "directly" above it. For example, in row 5: __1=1__ ..... __(1+4)=5__ ..... __(4+6)=10__ ..... __(4+6)=10__ ..... __(4+1)=5__ ..... __1=1__
 * We can see that every line is symmetrical and starts and ends with 1.

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__//Road Junction!//__

To continue learning on the Pascal's Triangle, click on Pascal's Triangle in the sidebar.

To learn about the binomial theorem, click on Binomial's Theorem in the sidebar.

To leap straight into another form of geometrical arrangement of numbers also in a triangle, which has similarities and differences with the Pascal's triangle, click on Leibniz Harmonic Triangle

Go on and click a button to start exploring more!