Thursday, January 17, 2013

The binomial coefficients

The translation of the word binomial means "two names". In algebra, a polynomial is an expression with several terms, a term being a set of numbers and variables multiplied together. The number in the term is called the coefficient.  Here are some examples

7xyz² is a single term with three variables. 7 is the coefficient, Since the z is raised to the second power, if we wrote this out in letters it would be 7xyzz, and since we have to used four letters to write this out completely, we say this is a fourth degree term.

7xy - 2z² is two terms, so it is called a binomial. The coefficient of the first term is 7 and the coefficient of the second term is -2.

What happens when we raise a binomial to different powers? Here are the first few examples.

(x + y)ยบ = 1
(x + y)¹ = x + y
(x + y)² = x² + 2xy + y²
(x + y)³ = x³ + 3x²y + 3xy² + y³

If we strip away the variables and just look at the coefficients, we get

1 1
1 2 1
1 3 3 1

If you have been reading the blog this week, you know those are the first few rows of Pascal's triangle.

Here is the standard summation form of this fact, usually called The Binomial Theorem.

Notice that the powers for x and y always add up to n when we raise (x + y) to the n-th power.

You might also notice that the sums of the rows of Pascal's Triangle has an easily recognizable pattern.

1 = 1
2 = 1+1
4 = 1+2+1
8 = 1+3+3+1

In these first few cases, it looks like the sum of the n-th row is 2 raised to the power of n. There are many ways to prove this pattern continues forever. 

Tomorrow, we will use The Binomial Theorem to prove this pattern.

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