Solving the Problem!

**By Pranav , Nishka**

**THE BINOMIAL THEOREM**

The Easy Stuff

Application

Binomial Expansion describes

the algebraic powers of a binomial.

In other words it is possible to

(x + y)^n into a sum. (E.G. ax^b y^c)

This formula and the triangle was

created by Blaise Pascal in the 17th century.

Pascal was most commonly known as a mathematician, physicist, and inventor.

Okay So lets get acquainted.

The sum of a+b is called a binomial (It contains 2 terms).

Any expression like (a+b)^n is called a power of a binomial.

We Can say that a^2 +2ab+b^2 is the binomial expansion of (a+b)^2

or

a^3 +3a^2 b+3ab^3+b^3 is the binomial expansion of (a+b)^3

So Your now probably thinking, well this is nothing hard. Since its basically like multiplying factors.

Your right, although how are you gonna solve something that is to the 4th or 5th power? Or even greater? Not so easy now is it.

One real life application of binomial theorem is in computing. It is useful for the distribution of IP addresses. With binomial theorem, the automatic distribution of IP addresses is not only possible but also the distribution of virtual IP addresses.

Another field that used Binomial Theorem as the important tools is the nation’s economic prediction. Economists used binomial theorem to count probabilities that depend on numerous and very distributed variables to predict the way the economy will behave in the next few years. To be able to come up with realistic predictions, binomial theorem is used in this field.

Now that we have learnt the concepts let's see how GDC makes our lives easier.

Okay now that we know the base equation for (a+b)^4 lets apply to a problem.

Lets Expand (1+2x)^4

1. The expression a^4+4a^b+ 6a^2 b^2+4ab^3+b^4

2. State your a and b a=1 b=2x

3. Substitute your a and b into the expression

4. (1)^4 +4(1)^3 (2x) + 6(1)^2 (2x)^2 + 4(1)(2x)^3 + (2x)^4

5. 1+ (4)(1)(2x) + (6)(1)(4x^2) + (4)(1)(8x^3) + 16x^4

6. So... 1+8x+24x^2+32x^3+16x^4

YOU DID ITTT!!!!!

Here's a quick video.

Using the GDC is an easier option for gathering all the coefficients rather than manually working it out with the formula.

nCr can be found by table where C is available in options->prob->c.

Thank you !!!

So then how do we the formula part of the binomial theorem?

We Have to use this:

We Call this the factorial equation.

What does this mean? Well....

Then what is n and k? Lets look at our patterns reference again.. If the power was 4, that is your nth term! In other words the n stands for: The first terms exponent! So x^4 the nth term is 4

What is k? well its your second term's exponent. Like in our example we rendered that the nth term is 4, so the K will eventually be 4! So the first k is 0, then 1 then 2 then 3 and lastly 4.

PUTTIN' IT ALL TOGETHER

Well it looks like a jigsaw isnt it? Well it is...

Lets look at an example!

So, the n is 3. If your wondering the E is called a Sigma. On top of the sigma is the n, which in this case is 3. We also know our k will always start of as 0. In the brackets it's your n on top and k on the bottom. Then whats all the math next to it?

ITS OUR FACTORALS!! So it should start making sense now, you solve the factorals, and you should get the coefficient, which then you multiply by the factors and BBBAAAAAAMMMMM!!

You Get Your Expansion! :]

Were done!

Yay! Grats on learning the complex Binomial Expansion! We would now like to take any questions. But first here's a quick problem C;

THANK YOU !!!