Editing Reciprocal polynomial (section)

=={{anchor|Palindromic polynomial|Antipalindromic polynomial}} Palindromic and antipalindromic polynomials==
A self-reciprocal polynomial is also called palindromic because its coefficients, when the polynomial is written in the order of ascending or descending powers, form a [[palindrome]]. That is, if
:<math> P(x) = \sum_{i=0}^n a_ix^i</math> 
is a polynomial of [[Degree of a polynomial|degree]] {{math|''n''}}, then {{math|''P''}} is ''palindromic'' if {{math|1=''a<sub>i</sub>'' = ''a''<sub>''n''−''i''</sub>}} for {{math|1=''i'' = 0, 1, ..., ''n''}}. 

Similarly, a polynomial {{math|''P''}} of degree {{math|''n''}} is called '''antipalindromic''' if {{math|1=''a<sub>i</sub>'' = −''a''<sub>''n''−''i''</sub>}} for {{math|1=''i'' = 0, 1, ..., ''n''}}. That is, a polynomial {{math|''P''}} is ''antipalindromic'' if {{math|1=''P''(''x'') = –''P''<sup>∗</sup>(''x'')}}.

===Examples===
From the properties of the [[binomial coefficient]]s, it follows that the polynomials {{math|1=''P''(''x'') = (''x'' + 1)<sup>''n''</sup>}} are palindromic for all positive [[integer]]s {{math|''n''}}, while the polynomials {{math|1=''Q''(''x'') = (''x'' – 1)<sup>''n''</sup>}} are palindromic when {{math|''n''}} is even and antipalindromic when {{math|''n''}} is [[parity (mathematics)|odd]].

Other examples of palindromic polynomials include [[cyclotomic polynomial]]s and [[Eulerian polynomial]]s.

===Properties===
* If {{math|''a''}} is a root of a polynomial that is either palindromic or antipalindromic, then {{sfrac|{{math|''a''}}}} is also a root and has the same [[multiplicity (mathematics)|multiplicity]].<ref>{{harvnb|Pless|1990|loc=pg. 57}} for the palindromic case only</ref>
* The converse is true: If for each root {{math|''a''}} of polynomial, the value {{sfrac|{{math|''a''}}}} is also a root of the same multiplicity, then the polynomial is either palindromic or antipalindromic.
* For any polynomial {{math|''q''}}, the polynomial {{math|''q'' + ''q''<sup>∗</sup>}} is palindromic and the polynomial {{math|''q'' − ''q''<sup>∗</sup>}} is antipalindromic.
* It follows that any polynomial {{math|''q''}} can be written as the sum of a palindromic and an antipalindromic polynomial, since {{math|1=''q'' = (''q'' + ''q''<sup>∗</sup>)/2 + (''q'' − ''q''<sup>∗</sup>)/2}}.<ref>{{citation|first=Jonathan Y.|last=Stein|title=Digital Signal Processing: A Computer Science Perspective|publisher=Wiley Interscience|year=2000|page=384|isbn=9780471295464}}</ref>
* 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 {{math|''x'' + 1}} (it has –1 as a root) and its quotient by {{math|''x'' + 1}} is also palindromic.
* An antipalindromic polynomial over a field {{mvar|k}} with odd [[Characteristic (algebra)|characteristic]] is a multiple of {{math|''x'' – 1}} (it has 1 as a root) and its quotient by {{math|''x'' – 1}} is palindromic.
* An antipalindromic polynomial of even degree is a multiple of {{math|''x''<sup>2</sup> – 1}} (it has −1 and 1 as roots) and its quotient by {{math|''x''<sup>2</sup> – 1}} is palindromic. 
* If {{math|''p''(''x'')}} is a palindromic polynomial of even degree 2{{mvar|d}}, then there is a polynomial {{math|''q''}} of degree {{math|''d''}} such that {{math|1=''p''(''x'') = ''x''<sup>''d''</sup>''q''(''x'' + {{sfrac|1|''x''}})}}.<ref>{{harvnb|Durand|1961}}</ref>
* If {{math|''p''(''x'')}} is a [[monic polynomial|monic]] antipalindromic polynomial of even degree 2{{mvar|d}} over a field {{mvar|k}} of odd [[Characteristic (algebra)|characteristic]], then it can be written uniquely as {{math|1=''p''(''x'') = ''x''<sup>''d''</sup>(''Q''(''x'') − ''Q''({{sfrac|''x''}}))}}, where {{mvar|Q}} is a monic polynomial of degree {{mvar|d}} with no constant term.<ref>{{citation|first=Nicholas M.|last=Katz|title=Convolution and Equidistribution : Sato-Tate Theorems for Finite Field Mellin Transformations|publisher=Princeton University Press|year=2012|isbn=9780691153315|page=146}}</ref>
* If an antipalindromic polynomial {{math|''P''}} has even degree {{math|2''n''}} over a field {{mvar|k}} of odd characteristic, then its "middle" coefficient (of power {{math|''n''}}) is 0 since {{math|1=''a''<sub>''n''</sub> = −''a''<sub>2''n''&thinsp;–&thinsp;''n''</sub>}}.

===Real coefficients===
A polynomial with [[Real number|real]] coefficients all of whose [[Complex number|complex]] roots lie on the unit circle in the [[complex plane]] (that is, all the roots have modulus 1) is either palindromic or antipalindromic.<ref>{{cite book|first1=Ivan|last1=Markovsky|first2=Shodhan|last2=Rao|title=2008 16th Mediterranean Conference on Control and Automation |chapter=Palindromic polynomials, time-reversible systems, and conserved quantities |publisher=IEEE|pages=125–130|year=2008|doi=10.1109/MED.2008.4602018|isbn=978-1-4244-2504-4|s2cid=14122451 |url=https://eprints.soton.ac.uk/266592/1/Med08.pdf}}</ref>