. Binomial Theorem Question/Help .

[xLaser]

Hello Guys, I just have a few quick question on binomial theorem, any help would be greatly appreciated.

1. Expand using the binomial theorem in powers of x up to and including x^3 : (1 + 2x + 3x^2)^5 : I always thought binomial theorem would be used to expand binomials... this is not a binomial unless i do htis [(1+2x) + (3x^2)^5]... now the problem comes when I simplifiy and collect like-terms because there will be a (1+2x)^5 and that is difficult to expand, is there a easier way or am I messing up somewhere?

2. General question here, "is there any real life uses for the binomial theorem and what the limitations... When shouldn't/can't you use it?"

this may seem obvious but after all binomial theorem are used to expand bionomials... that is one limitations, are there others or any real l ife use? please advice, thanks a lot guys.

 

[dextercioby]

The second part is phylosophical and is none of my interest.

Apparently,you know what to do.You must apply the theorem twice,so do it.

Daniel.

 

[xLaser]

dextercioby ok i expanded the whole thing by applying the theorem several times, but it is the 2nd part of the question that is confusing me... xpand using the binomial theorem in powers of x up to and including x^3 ... i interpted this as expanding only to the point where x is cubed... but how do i know when that is going to happen, maybe the question do not want a full expansion? please assist, thx.

 

[Dr.Brain]

Yes, as per your second question, you must write only the terms upto $x^3$, one more thing, can you do something like write $(1+2x+3x^2)$ as a square of a linear?... try it , it will reduce your work.

 

[xLaser]

Brain can't factor that... but i guess to write up to x^3 i sitll have to expand everything but for the final answre i only write terms up to x^3?

 

[Dr.Brain]

yes...first expand it completely...then sort the terms in such a way that u end up arranging the terms in increasing powers of x , then remove others and write till x^3 . that will be your answer.

 

[xLaser]

ok collected like terms till X^3

it became 1+10x+24x^2-40x^3

 

[Hurkyl]

"Real life" uses of the binomial theorem? Ok, I'm a mathematician / computer scientist, so I don't have a "real life" , but I'll answer anyways.

I've used it in two main ways:

(1) To quickly expand a binomial raised to a power. Saves a lot of arithmetic, thus reducing the likelyhood I'll make a silly mistake. The idea generalizes to multinomials, but by that point it would be quicker to load up a computer algebra program!

(2) To factor! This is arguably more important than expanding. There are all sorts of messy sums one will often encounter in the wild, and the binomial theorem let's you tame some of them into something much simpler.

 

[Dr.Brain]

Actually most of the mathematics does not involve direct-use, but it may find its use in physics and thus is indirectly helping us in solving some real life problems.

 

[xLaser]

ok is there an example in physics where they use binomial theorem?

 

[xLaser]

um guys 1 last question: can't seem to get this one properly.

Q: The constant terms (this is when x ^ 0) in the expansions of (px^3 + q/x^3)^8 and (px^2 + q/x^2)^4 are equal. Both p and q are grater htan zero. Express p in terms of q.

wut i did was write out the general term for both expansions, and then found the value of the constant term. which turns out of be (8 choose 4)p^4q^4 = (4 choose 2)p^2q^2 ... now this might be wrong, i think it is wrong because how do u express p in terms of q like this?

please advice, thx.

 

[Dr.Brain]

ok is there an example in physics where they use binomial theorem?

Ok let's see , Ihave used binomial in gravitation for variation of g with height and depth when the height/depth is small as compared to Earth's radius,I have used it in shell theorems and numerous general physics problems.