Binomial Theorem

[onur369]

Can someone please explain it in year 12 standards. Im not sure what exactly I have to know about the Binomial Theorem but Im finding it hard to understand. Can you genius' explain to me exactly what this theorem is?

[brightsky]

Basically just Pascal's triangle, it's used for expansion. Just say you have something like (ax + b)^6. Then the binomial theorem helps you expand this in one step without the hassle of expanding term by term. Essentially it just utilises combinations (how many of a certain term there is) and hence figures out the coefficient in front of it - the expansion is as follows . The theory behind the derivation of the theorem is quite straight forward. If you observe the Pascal's triangle, you'll notice the pattern (the pattern exists because of the way it is expanded).

[Greatness]

It pretty much allows you to expand

[aznxD]

Pretty much this:


The coefficients can be observed in pascal's triangle. The power of the first term decreases by one, and the second increases by one.

For example:

[werdna]

Oh man I'm so glad I'm not repeating this subject.

[Greatness]

aznxD and brightsky have got it covered. but with pascals triangle you will notice that the 6th row is 1 5 10 10 5 1 thus giving the coefficents of the x terms in descending order.

[onur369]

So I just opened the book for the first time. Example 1, in relation to the Binomial Theorem. Expand (2x-3)^4 using the Binomial Theorem. Ok I do the first step properly, but then I dont know what do to on the other part. thats the final answer but i dont know what do the step before the answer.

I do all this correct but its the step right afterwards.

[Greatness]

This part (4/0)x(2x)^4becomes 16x^4. If you look at pascals triangle you will notice that in the 5th row, the first term is 1 so you multiply 16x^4 by 1 which gives you 16x^4. You do that with each term, but you would find all the x terms first then look at the pascals triangle to determine the rest of the coefficients.
          I need to learn how to use latex...

[aznxD]

You got to substitute it with the right numbers.

It's:




 
This becomes:

EDIT: I also got to learn how to use latex efficiently. This took me like 15 mins -_-

[onur369]

 
This becomes:

I dont understand how we get 16x^4 :@

[Greatness]

(2x)^4=16x^4

[aznxD]

Do you know how to work out combinations?
For the first term:

[onur369]

Woops, I actually ment the other part,  2x^3 (-3)^1= -24x^3. How do we get -96x^3?

[Greatness]

Pascals triangle:
                         1
                       1   1
                     1   2   1
                   1   3   3   1
                 1   4   6   4   1   <<<<<<<<< This is the row you are looking at. Multiply each x term by these to give the coefficients
              

[Greatness]

you multiply the -24x^3 by 4 as you can see from pascals triangle. this is a pretty simple method - that is pascals
x^4 term gets multplied by 1
x^3   ''       ''       ''        ''  4
x^2   ''       ''       ''        ''  6
x^1   ''       ''       ''        ''  4
x^0   ''       ''       ''        ''  1