Editing Reciprocal polynomial (section)

===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>}}.