Proving the General Formula for X^n - y^n (2025)

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.

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

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.

[tex]
(x-y)(x^{n-1} + x^{n-2}y + ... + xy^{n-2} + y^{n-1})
[/tex]

[tex]
(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})
[/tex]

[tex]
(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})
[/tex]

[tex]
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}
[/tex]

[tex]
x^{n} + ... + x^{2}y^{n-2} - x^{n-2}y^{2} - ... - y^{n}
[/tex]

... now I'm stuck.

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

[tex]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})[/tex]

The first factor is expanded:
[tex]x^n+x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1}[/tex]

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

Do you notice any cancelling pattern happening?

Mentallic said:

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

[tex]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})[/tex]

The first factor is expanded:
[tex]x^n+x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1}[/tex]

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

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.

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 [tex]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})[/tex]

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.

[tex]x^n+x^{n-1}y+x^{n-2}y^2+...+x^2y^{n-2}+xy^{n-1}[/tex]
[tex]...-x^{n-1}y-x^{n-2}y^2-... -x^2y^{n-2} - xy^{n-1} - y^n[/tex]

get the idea?