Understanding the Ellipsis in Algebraic Operations

[humanaction]

Homework Statement

Prove xⁿ-yⁿ=(x-y)(x^n-1+x^n-2y+…+xy^n-2+y^n-1

Homework Equations

12 basic properties of numbers

The Attempt at a Solution

I'm all good except I don't see why x²y^n-2-x^n-2y²=0

 

[Ray Vickson]

Homework Statement

Prove xⁿ-yⁿ=(x-y)(x^n-1+x^n-2y+…+xy^n-2+y^n-1

Homework Equations

12 basic properties of numbers

The Attempt at a Solution

I'm all good except I don't see why x²y^n-2-x^n-2y²=0

This is not true unless x = y (or sometimes x = -y as well, depending on n). Anyway, why do you write this? It has nothing at all to do with the original question.

 

[humanaction]

This is not true unless x = y (or sometimes x = -y as well, depending on n). Anyway, why do you write this? It has nothing at all to do with the original question.

How does it not apply? x(xy^n-2)-y(x^n-2y)= aforementioned dilemma, ignoring the rest of the equation because I'm on my phone...

 

[SteamKing]

You probably should not try to do complicated math on your phone.

 

[haruspex]

Homework Statement

Prove xⁿ-yⁿ=(x-y)(x^n-1+x^n-2y+…+xy^n-2+y^n-1

Homework Equations

12 basic properties of numbers

The Attempt at a Solution

I'm all good except I don't see why x²y^n-2-x^n-2y²=0

Are you perhaps confusing (x^n-1+x^n-2y+…+xy^n-2+y^n-1) with (x^n-1+x^n-2y+xy^n-2+y^n-1)?
The ellipsis ("+...+") is important.

 

[humanaction]

ah yes. so multiplication works the same it seems, but how do you subtract ellipses? sorry I've been out of school for 5 years and never went to uni :(

 

[VantagePoint72]

The ellipses means "plus everything else that follows that is pattern". i.e. 1+2+3+...+10 = 1+2+3+4+5+6+7+8+9+10. It's just short hand to avoid writing out many terms when their form is (supposedly) obvious from the terms given.

 

[VantagePoint72]

I know you're just asking for help with this one problem, but I'm going to offer some unsolicited academic advice: Spivak's calculus is a very rigorous book and might not be the best choice for teaching yourself calc if you've been out of school for a long time. Since the first part is meant to be review, it probably doesn't bode well if you're getting stuck on the very first problem. Usually one's first encounter with calculus should be a bit more informal than Spivak's treatment to get an intuitive understanding of foreign concepts like limits before learning how to do it in full mathematical rigour. In any case, it sounds like you need to review some high school math like algebra and trig before tackling calculus.

 

[humanaction]

i agree. i was going to ask what is the best precalculus book? i am planning on going to uni next autumn and i would like to be able to test out of all my maths if possible to save money. so time is limited. but at the same time i want to learn it right. i enjoy maths and am not just trying to pass classes, i want to learn the material. thank you for your help.

 

[VantagePoint72]

I can't recommend a particular pre-calculus book from my own experience, but these: http://math.about.com/od/booksresourcesdvds/tp/algebra1.htm all look solid for algebra. I've used other books in some of those series with success, so I think the recommendations are probably good.

That said, and while it's certainly a great idea to do some advance work prior to starting uni, attempting to test out of all your math is probably a bit overly ambitious. Plus, it might not even make a difference to your time and cost: typically, universities require you do a certain number of credits (with what exactly constitutes a credit varying by institution) to graduate. With the exception of certain accredited high school programs like AP and the international baccalaureate, testing out of a class rarely means you get the credit for it. It just means you get to skip the prerequisite. You would still need to complete the same number of credits to graduate. I'm guessing you're from the UK, based on a few of your word choices? I'm most familiar with the North American system, but I'm a graduate student at a UK institution right now and I think the situation is unlikely to be different. At any rate, you should check with your prospective institution to see what their policy is. To be honest, it's probably in your best interest to just take the classes at the school. University level mathematics can be quite challenging, and having the support of TAs, your professor, and classmates who are working through the same material can make a world of difference.

 

[humanaction]

thank you for your information. my uni allows me to test out of and earn credit for up to calculus iii. i will look into those books. in the meantime, could you try to explain this ellipsis to me? i haven't run into any problems with the other questions from this chapter, I've just never done an algebraic operation of an ellipsis before.

 

[Dick]

thank you for your information. my uni allows me to test out of and earn credit for up to calculus iii. i will look into those books. in the meantime, could you try to explain this ellipsis to me? i haven't run into any problems with the other questions from this chapter, I've just never done an algebraic operation of an ellipsis before.

(x^3-y^3)=(x-y)(x^2+xy+y^2)=(x-y)(x^2+...+y^2). In that case the ... stands for xy. (x^4-y^4)=(x-y)(x^3+xy^2+x^2y+y^3)=(x-y)(x^3+...+y^3). In that case the ... stands for xy^2+x^2y. It just stands for everything they left out in the expression. You can't do an algebraic operation on it. You just have to figure out what it means.