Post History
#2: Post edited
by
trichoplax
·
2023-08-22T14:18:22Z (over 1 year ago)
Replace excessively long MathJax title so that it fits on the page
Picture proof for $x^n−y^n=(x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$
- Picture proof for expansion of $x^n−y^n$
Most students fail to intuit [$x^n−y^n \equiv (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$ ](https://math.stackexchange.com/a/117676) as substantiated by the glut of duplicates, at least 20 on Math StackExchange. Thus how can students pictorialize it? I seek **solely **VISUAL (not algebraic) proofs here.
After substituting $z = \dfrac xy$, above identity follows from $z^n−1 \equiv (z−1)(z^{n−1}+z^{n−2}+...+z+1)$.
I couldn't find a picture proof from Roger B. Nelson's [_Proofs without Words_ (1993),
_Proofs without Words II_ (2000), or
_Proofs without Words III_ (2015)](https://math.stackexchange.com/a/3457754).
#1: Initial revision
by
Este
·
2023-08-22T07:31:45Z (over 1 year ago)
Picture proof for $x^n−y^n=(x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$
Most students fail to intuit [$x^n−y^n \equiv (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$ ](https://math.stackexchange.com/a/117676) as substantiated by the glut of duplicates, at least 20 on Math StackExchange. Thus how can students pictorialize it? I seek **solely **VISUAL (not algebraic) proofs here.
After substituting $z = \dfrac xy$, above identity follows from $z^n−1 \equiv (z−1)(z^{n−1}+z^{n−2}+...+z+1)$.
I couldn't find a picture proof from Roger B. Nelson's [_Proofs without Words_ (1993),
_Proofs without Words II_ (2000), or
_Proofs without Words III_ (2015)](https://math.stackexchange.com/a/3457754).