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

#Calculus

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Comments

See Chebyshev Polynomials.

Pi Han Goh - 9 months, 1 week ago

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

Elijah L - 9 months, 1 week ago

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

Pi Han Goh - 9 months, 1 week ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$