Proving the General Formula for X^n - y^n

[DavidSnider]

Homework Statement

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

Homework Equations

See Above

The Attempt at a Solution

The previous problem in the book was:
Prove:
x³ - y³ = (x - y)(x² + xy + y²)

(x - y)(x² + xy + y²)
(x)(x² + xy + y^²) + (-y)(x² + xy + y²)
(x³ + x²y + xy²) + (-x²y - xy² - y³)
x³ + x²y + xy² - x²y - xy² - y³
x³ - y³

I'm not sure how to show the same thing when the exponent is variable though.

 

[Dick]

Do the same thing you did for x^3-y^3. Multiply out the right side. Many terms cancel.

 

[DavidSnider]

I know they cancel, but.. how am I supposed to show that?

 

[pyrosilver]

random do you happen to be using the Spivaks calculus book?

try multiplying (x-y) with all of the terms you have listedm like xn-1, xn-2y, etc and go from there

 

[DavidSnider]

Yes, I am using Spivak's calculus. It's unlike any math book I've ever used before, so I am kind of confused as to what they are expecting me to do.

I'll think about what you just said.

 

[pyrosilver]

I definitely agree with you. I too am using Spivak's calculus book (my class just finished chapter 2, I'm a sophomore so I'm going a little slower through the book). But yeah start by multiplying the beginning terms you have, and the end terms you have. good luck!

 

[Dick]

Write out x*(x^(n-1)+x^(n-2)*y+...+x*y^(n-2)+y^(n-1)) and y*(x^(n-1)+x^(n-2)*y+...+x*y^(n-2)+y^(n-1)) and look for things that cancel. E.g. x*x^(n-2)*y cancels y*x^(n-1), x*x^(n-3)*y^2 cancels y*x^(n-2)*y. I know you can't write out all of the terms. You'll have to use the '...' to express what you mean. It might help to write the two expanded products on separate lines and shift one over so cancelling terms are above each other.

 

[DavidSnider]

$$(x-y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1})$$

$$(x)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1}) + (-y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1})$$

$$(x^{n-1+1} + x^{n-2+1}y + ... + x^{2}y^{n-2} + xy^{n-1}) + (-x^{n-1}y - x^{n-2}y^{2} - ... - xy^{n-2+1} - y^{n-1+1})$$

$$x^{n} + x^{n-1}y + ... + x^{2}y^{n-2} + xy^{n-1} -x^{n-1}y - x^{n-2}y^{2} - ... - xy^{n-1} - y^{n}$$

$$x^{n} + ... + x^{2}y^{n-2} - x^{n-2}y^{2} - ... - y^{n}$$

... now I'm stuck.

 

[Mentallic]

Well, to prove $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+...+xy^{n-2}+y^{n-1})$$
we are just going to expand the RHS.

$$RHS=x(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1}) - y(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})$$

The first factor is expanded:
$$x^n+x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1}$$

The second factor is expanded:
$$-(x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1}+y^n)$$

Do you notice any cancelling pattern happening?

 

[DavidSnider]

Well, to prove $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+x^{n-3}y^2+...+xy^{n-2}+y^{n-1})$$
we are just going to expand the RHS.

$$RHS=x(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1}) - y(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})$$

The first factor is expanded:
$$x^n+x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1}$$

The second factor is expanded:
$$-(x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1}+y^n)$$

Do you notice any cancelling pattern happening?

Yeah I knew they all canceled intuitively, just wasn't sure how to show it on paper.
Filling in the blanks one step further makes it more clear.

 

[Mentallic]

Well how about making it obvious to the examiner that you realize they cancel by lining up each equal term?

i.e. after the line $$x(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1}) - y(x^{n-1}+x^{n-2}y+...+xy^{n-2}+y^{n-1})$$

Then expand the first factor on 1 line, then expand the next factor on the line underneath, but keep cancelling factors in line with each other.

$$x^n+x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1}$$
$$...-x^{n-1}y-x^{n-2}y^2-... -x^2y^{n-2} - xy^{n-1} - y^n$$

get the idea?