Editing Chebyshev polynomials (section)

===Roots and extrema===
A Chebyshev polynomial of either kind with degree {{mvar|n}} has {{mvar|n}} different [[simple root]]s, called '''Chebyshev roots''', in the interval {{closed-closed|−1, 1}}. The roots of the Chebyshev polynomial of the first kind are sometimes called [[Chebyshev nodes]] because they are used as ''nodes'' in polynomial interpolation. Using the trigonometric definition and the fact that:
<math display="block">\cos\left((2k+1)\frac{\pi}{2}\right)=0</math>
one can show that the roots of {{mvar|T<sub>n</sub>}} are:
<math display="block"> x_k = \cos\left(\frac{\pi(k+1/2)}{n}\right),\quad k=0,\ldots,n-1.</math>
Similarly, the roots of {{mvar|U<sub>n</sub>}} are:
<math display="block"> x_k = \cos\left(\frac{k}{n+1}\pi\right),\quad k=1,\ldots,n.</math>
The [[Maxima and minima|extrema]] of {{mvar|T<sub>n</sub>}} on the interval {{math|−1 ≤ ''x'' ≤ 1}} are located at:
<math display="block"> x_k = \cos\left(\frac{k}{n}\pi\right),\quad k=0,\ldots,n.</math>

One unique property of the Chebyshev polynomials of the first kind is that on the interval {{math|−1 ≤ ''x'' ≤ 1}} all of the [[Maxima and minima|extrema]] have values that are either −1 or 1. Thus these polynomials have only two finite [[Critical value (critical point)|critical value]]s, the defining property of [[Shabat polynomial]]s. Both the first and second kinds of Chebyshev polynomial have extrema at the endpoints, given by:
<math display="block">\begin{align}
T_n(1) &= 1 \\
T_n(-1) &= (-1)^n \\
U_n(1) &= n+1 \\
U_n(-1) &= (-1)^n (n+1).
\end{align}</math>

The [[Maxima and minima|extrema]] of <math>T_n(x)</math> on the interval <math>-1 \leq x \leq 1</math> where <math>n>0</math> are located at <math>n+1</math> values of <math>x</math>. They are <math> \pm 1</math>, or <math>  \cos\left(\frac{2\pi k}{d}\right)</math> where <math>d > 2</math>, <math>d \;|\; 2n</math>, <math>0 < k < d/2</math> and <math>(k, d) = 1</math>, i.e., <math>k</math> and <math>d</math> are relatively prime numbers.

Specifically ([[Minimal polynomial of 2cos(2pi/n)]]<ref name=Gurtas>{{cite journal |first1=Y. Z. |last1=Gürtaş |title=Chebyshev Polynomials and the minimal polynomial of <math>\cos (2 \pi/n)</math> |year=2017 |journal=American Mathematical Monthly |volume=124 |number=1 |pages=74–78 |doi=10.4169/amer.math.monthly.124.1.74|s2cid=125797961 }}</ref><ref name=Wolfram0>{{cite journal |first1=D. A. |last1=Wolfram |title=Factoring Chebyshev polynomials of the first and second kinds with minimal polynomials of <math>\cos (2 \pi /d )</math> |year=2022 |journal=American Mathematical Monthly |volume=129 |number=2 |pages=172–176 |doi=10.1080/00029890.2022.2005391|s2cid=245808448 }}</ref>) when <math>n</math> is even: 
* <math>T_n(x) = 1</math> if <math>x = \pm 1</math>, or <math>d > 2</math> and <math>2n/d</math> is even. There are <math>n/2 + 1</math> such values of <math>x</math>.
* <math>T_n(x) = -1</math> if  <math>d > 2</math> and <math>2n/d</math> is odd. There are <math>n/2</math> such values of <math>x</math>.

When <math>n</math> is odd: 
* <math>T_n(x) = 1</math> if <math>x = 1</math>, or <math>d > 2</math> and <math>2n/d</math> is even. There are <math>(n+1)/2</math> such values of <math>x</math>.
* <math>T_n(x) = -1</math> if  <math>x = -1</math>, or <math>d > 2</math> and <math>2n/d</math> is odd. There are <math>(n+1)/2</math> such values of <math>x</math>.