# nth degree polynomials inside square. $y=4x^3-3x$ $y=8x^4 - 8x^2 +1$

The two pictures are examples of third and fourth degree polynomials 'just fitting' inside a square of side length 2 and centered about the origin. These surprisingly have integer coefficients.

What can be said about polynomials of a higher degree?

Here are degrees 2-5:

$2x^2-1$

$4x^3-3x$

$8x^4-8x^2+1$

$16x^5-20x^3+5x$ (By trial and error)

It is easy to show that the coefficients must sum to 1 and it appears the leading coefficient increases by powers of 2 Note by Chris Sapiano
2 weeks, 4 days ago

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- 2 weeks, 3 days ago

Interesting that there is this connection, why is this so?

- 2 weeks, 3 days ago

I guess it's just a corollary of Chebyshev polynomials. I can't verbally explain this.

- 2 weeks, 3 days ago