How to picture $x^n−y^n \equiv (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$?

Question

Many students fail to intuit $x^n−y^n \equiv (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$ , as substantiated by the glut of duplicates. How can students pictorialize it?

After substituting $z = \dfrac xy$, this identity follows from $z^n−1 \equiv (z−1)(z^{n−1}+z^{n−2}+...+z+1)$.

I need to motivate this identity from Michael Spivak, Calculus (4 edn 2008), p. 13, Problem 1(v). But I found no picture proof for this in Roger B. Nelson's Proofs without Words (1993),
Proofs without Words II (2000), or Proofs without Words III (2015).

$z^n−1 = (z−1)(z^{n−1}+z^{n−2}+...+z+1)$ is basically partial sum of geometric series, if that helps. — Ennar, Sep 17 '23 at 22:45
I understand that you know it, but I'm suggesting looking at this through lens of geometric sequences instead of "random" algebraic identity. — Ennar, Sep 17 '23 at 22:49
For $n=2,3$, it can be visualized with differences of areas/volumes. — Cameron Williams, Sep 18 '23 at 00:49
There is this picture proof by Roger Nelson: https://maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/proof-without-words-geometric-series — MathFont, Sep 18 '23 at 01:00
@MathSensei That PDF pictorializes Geometric Series, not this identity. — , Sep 18 '23 at 08:30
@CameronWilliams Thanks. I know. Can this identity $\forall ; n \ge 4$ be visualized with differences of HyperAreas of HyperVolumes? — , Sep 18 '23 at 23:52

Rob Arthan · Accepted Answer · 2023-11-01T02:14:13.790

How about using a tabular format to display the result of expanding something of the form $(x - y)z$ as $xz -yz$? Thus: $$ (x−y)(x^{n−1}+x^{n−2}y+{}...{}+xy^{n−2}+y^{n−1}) ={}\\ \\ \begin{array}{c@{}c@{}c@{}c@{}c@{}c@{}c@{}c@{}} &[(x^n + {}&x^{n-1}y + {}& x^{n-2}y^2 + {}& \ldots + {}&x^2y^{n-2} + {} & xy^{n-1}) & {} - {} &\\ &&(x^{n-1}y {}+{}& x^{n-2}y^2 +{}& \ldots + {} & x^2y^{n-2} + {}& xy^{n-1} + {}& y^n)]&\\ =&x^n + {}&0+{}&0+{}&\ldots+{}&0+{}&0 &{}-y^n&\\ =& x^n - y^n \end{array} $$ That's my picture of how the algebra works.

How to picture $x^n−y^n \equiv (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$?

1 Answers1