square root of polynomial

The   f,  denoted by f, is any polynomialPlanetmathPlanetmath g having the square  g2 equal to f.  For example,  9x2-30x+25=3x-5 or -3x+5.

A polynomial needs not have a square root, but if it has a square root g, then also the opposite polynomial -g is its square root.

AlgorithmMathworldPlanetmath.  The idea of the squaring


(see the square of sum) gives a method for getting the square root of a polynomial:

  • The .

  • And so on.

In the examples below, on the .

Example 1.9x4+6x3-11x2-4x+4= ?

(9x4 +6x3 -11x2 -4x +4) = ± (3x2 +x -2)
9x4 3x2
6x3 -11x2 6x2 +x
6x3 +x2 x
-12x2 -4x +4 6x2 +2x -2
-12x2 -4x +4 -2

Example 2.x6-2x5-x4+3x2+2x+1= ?

(1 +2x +3x2 -x4 -2x5 +x6) = ± (1 +x +x2 -x3)
1 1
2x +3x2 2 +x
2x +x2 +x
2x2 -x4 2 +2x +x2
2x2 +2x3 +x4 x2
-2x3 -2x4 -2x5 +x6 2 +2x +2x2 -x3
-2x3 -2x4 -2x5 +x6 -x3

Remark.  The procedure may give a Taylor series expansion of the square root, if it is not a polynomial.  E.g. we get



  • 1 Meyers Rechenduden.  Erster verbesserter Neudruck.  Bibliographisches Institut AG, Mannheim (1960).
Title square root of polynomial
Canonical name SquareRootOfPolynomial
Date of creation 2013-03-22 15:32:06
Last modified on 2013-03-22 15:32:06
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 17
Author pahio (2872)
Entry type Algorithm
Classification msc 26C99
Classification msc 12E05
Synonym calculation of square root of polynomial
Related topic SquareOfSum
Related topic BombellisMethodOfComputingSquareRoots