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.  $\sqrt{9x^4+6x^3-11x^2-4x+4}=$ ?

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

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

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