square root of polynomial

The   $f$,  denoted by $\sqrt{f}$, is any polynomial $g$ having the square  $g^2$ equal to $f$.  For example,  $\sqrt{9x^2-30x+25}=3x-5$ 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.

Algorithm.  The idea of the squaring

$${(a+b+c+..)}^2=(a)a+(2a+b)b+(2a+2b+c)c+..$$

(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.  $\sqrt{9x^4+6x^3-11x^2-4x+4}=$ ?

	$(9x^4$	$+6x^3$	$-11x^2$	$-4x$	$+4)$	$=$	$\pm$	$(3x^2$	$+x$	$-2)$
	$9x^4$							$3x^2$
		$6x^3$	$-11x^2$					$6x^2$	$+x$
		$6x^3$	$+x^2$						$x$
			$-12x^2$	$-4x$	$+4$			$6x^2$	$+2x$	$-2$
			$-12x^2$	$-4x$	$+4$					$-2$
					$0$

Example 2.  $\sqrt{x^6-2x^5-x^4+3x^2+2x+1}=$ ?

$(1$

$+2x$

$+3x^2$

$-x^4$

$-2x^5$

$+x^6)$

$=$

$\pm$

$(1$

$+x$

$+x^2$

$-x^3)$

$1$

$1$

$2x$

$+3x^2$

$2$

$+x$

$2x$

$+x^2$

$+x$

$2x^2$

$-x^4$

$2$

$+2x$

$+x^2$

$2x^2$

$+2x^3$

$+x^4$

$x^2$

$-2x^3$

$-2x^4$

$-2x^5$

$+x^6$

$2$

$+2x$

$+2x^2$

$-x^3$

$-2x^3$

$-2x^4$

$-2x^5$

$+x^6$

$-x^3$

$0$

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

$$\sqrt{1+x}=1+\frac{1}{2}x-\frac{1}{8}{x}^2+\frac{1}{16}{x}^3-\frac{5}{128}{x}^4+-\mathrm{\dots }$$

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