Complex Number Question.. Please help. (1 Viewer)

[Michaelmoo]

Hii there. Ive recently been studying Complex numbers straight from the Excel text book and a number of times; through the worked examples; large expansions are done in one line.

I dont understand how they happen; for example, (c +s)^6 is expanded in one line.

My question now, is the only way to do it by multiplying each (c +s) 6 times and collect like terms; (i.e. the text book is skipping a few steps) or is there a general formula???

i.e. what is the general formula for the expansion of (x +y)^n


More specifically, the question Im on is stating to express cos^7 theatre in terms of cosines of multiples of theatre (I can do it once I know the trick to the expansion).

Please help.

Thanks

 

[shaon0]

Hii there. Ive recently been studying Complex numbers straight from the Excel text book and a number of times; through the worked examples; large expansions are done in one line.

I dont understand how they happen; for example, (c +s)^6 is expanded in one line.

My question now, is the only way to do it by multiplying each (c +s) 6 times and collect like terms; (i.e. the text book is skipping a few steps) or is there a general formula???

i.e. what is the general formula for the expansion of (x +y)^n


More specifically, the question Im on is stating to express cos^7 theatre in terms of cosines of multiples of theatre (I can do it once I know the trick to the expansion).

Please help.

Thanks

Binomial theorem.
Example; (x+1)^3
= 3C0.x^3+3C1.x^2+3C2.x+3C3.x^0
= x^3+3x^2+3x+1

You can do the co-efficient expansion on your calculator.

 

[Michaelmoo]

Binomial theorem.

Binomial theorem??? Ive never come upon this?? Is it difficult to explain? Where can I find an online resource.

Thanks.

 

[shaon0]

Binomial theorem??? Ive never come upon this?? Is it difficult to explain? Where can I find an online resource.

Thanks.

Here is a resource that i downloaded, its quite brief but it wil help

 

[Michaelmoo]

Binomial theorem.
Example; (x+1)^3
= 3C0.x^3+3C1.x^2+3C2.x+3C3.x^0
= x^3+3x^2+3x+1

Wow thanks. that example explains it all, so basically you use the C function and take n at a time, n-1 at a time, n-2 at a time etc. And with the terms x and y, its simply X^n then x^n-1 times y1 ..... y^n. Thats Basically the pattern.

Thanks a lot!!!

 

[shaon0]

Wow thanks. that example explains it all, so basically you use the C function and take n at a time, n-1 at a time, n-2 at a time etc. And with the terms x and y, its simply X^n then x^n-1 x y1 ..... y^n. Thats Basically the pattern.

Thanks a lot!!!

Yeah, its from Pascal's triangle.

 

[lolokay]

using pascal's triangle (each number in a new row is the result of adding the 2 above it) to quickly get the different combinations can be quite helpful
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
etc. (arrange as an isoseles triangle)
This gives the coefficient for the expansion, with each new row representing the next power in the expansion (try and figure out why it works)

 

[Michaelmoo]

Also, does anyone have any resources of how to integrate for beginners (i.e. know absolutely nothing, although can differentiate). I know its not the first topic of Maths Extension 2 but the second part of the question asks me to find:

cos^7 theatre times d-theatre (with swigly line before, 0 at the bottom of the line and pi on two at the top).

Thanks.

 

[shaon0]

using pascal's triangle (each number in a new row is the result of adding the 2 above it) to quickly get the different combinations can be quite helpful
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
etc. (arrange as an isoseles triangle)
This gives the coefficient for the expansion, with each new row representing the next power in the expansion (try and figure out why it works)

Yeah thats how the calculator kind of does it.

 

[shaon0]

Also, does anyone have any resources of how to integrate for beginners (i.e. know absolutely nothing, although can differentiate). I know its notthe first topic but the second part of the question asks me to find:

cos^7 theatre times d-theatre (with swigly line before, 0 at the bottom of the line and pi on two at the top).

Thanks.

lol, don't worry about it.
I just did a question from fitzpatrick which was S(limits: pi/2,0) cosx^6 dx.

 

[lolokay]

Yeah thats how the calculator kind of does it.

..wouldn't the calculator just do n!/r!(n-r)! ?

 

[Michaelmoo]

Yeah thats how the calculator kind of does it.

Ok Thanks. Ill do the integration topic 1st and then come back to it.

 

[shaon0]

..wouldn't the calculator just do n!/r!(n-r)! ?

depends on the configuration but your most likely write.

 

[proringz]

Use DeMoivre's Theorem, otherwise use the general Binomial Expansion which you haven't learned at this stage. My advice would be to leave this part of complex numbers until you have learned Binomial Theorem in Extension 1.

 

[machiavel]

Also, does anyone have any resources of how to integrate for beginners (i.e. know absolutely nothing, although can differentiate). I know its not the first topic of Maths Extension 2 but the second part of the question asks me to find:

cos^7 theatre times d-theatre (with swigly line before, 0 at the bottom of the line and pi on two at the top).

Thanks.

The Arnold 4U text (you may know it as the Cambridge 4U) has an excellent integration page at the beginning of Chapter 5. Ignore the ax ones and memorise the f(x) ones. You can also treat integration as a Riemann sum if you really don't get it...

 

[adosh]

Re: HSC Tips - Harder 3U
<HR style="COLOR: #d1d1e1" SIZE=1><!-- / icon and title --><!-- message -->hello everyone,,, im having a problem wtih permuts combins,,,,now geha answers a questions as such "in how many ways can 8 people be arranged into two sets of four for tennis??" : (8C4x4C4)/2!= 35,,,now i understand this but heres a similar fitzpatrick question

find the number of wats in which 6 women and 6 men can be arranged in three sets of four for tennis wihcout restrictions??? and the asnwer is 155925.....now why cant we apply gehas method to this questuion, that is

(12C4x8C4x4C4)/3!=5775 but this is way off from 155925,,,,couls someone please help..thanks in advance