Properties
- If a is a root of a polynomial that is either palindromic or antipalindromic, then 1/a is also a root and has the same multiplicity.
- The converse is true: If a polynomial is such that "if a is a root then 1/a is also a root of the same multiplicity, then the polynomial is either palindromic or antipalindromic
- The product of two palindromic or antipalindromic polynomials is palindromic
- The product of a palindromic polynomial and an antipalindromic polynomial is antipalindromic
- A palindromic polynomial of odd degree is a multiple of x+1 (it has -1 as a root) and its quotient by x+1 is also palindromic
- An antipalindromic polynomial is a multiple of x-1 (it has 1 as a root) and its quotient by x-1 is palindromic
- An antipalindromic polynomial of even degree is a multiple of x2-1 (it has -1 and 1 as a roots) and its quotient by x2-1 is palindromic
- If p(x) is a palindromic polynomial of even degree 2d, then there is a polynomial q of degree d such that xdq(x+1/x) = p(x)
It results from these properties that the study the roots of a polynomial of degree d that is either palindromic or antipalindromic may be reduced to the study of the roots of a polynomial of degree at most d/2.
Read more about this topic: Palindromic Polynomial
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“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)