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Homesquare root of polynomial
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square root of polynomial
The square root of a polynomial $f$, 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 formula
$(a\!+\!b\!+\!c+..)^{2}=(a)a+(2a\!+\!b)b+(2a\!+\!2b\!+\!c)c+..$ 
(see the square of sum) gives a method for getting the square root of a polynomial:

The leading term of the root is equal to the square root of the first term of the radicand.

The second term of the root is equal to the first term of the first remainder divided by the double leading term.

The third term of the root is equal to first term of the second remainder divided by the double leading term.

And so on.
In the examples below, on the left under the lines there are the remainders, on the right under the lines the corresponding sums.
Example 1. $\sqrt{9x^{4}\!+\!6x^{3}\!\!11x^{2}\!\!4x\!+\!4}=$ ?
$\sqrt{}$  $(9x^{4}$  $+6x^{3}$  $11x^{2}$  $4x$  $+4)$  $=$  $\pm$  $(3x^{2}$  $+x$  $2)$  
$9x^{4}$  $3x^{2}$  
$6x^{3}$  $11x^{2}$  $6x^{2}$  $+x$  
$6x^{3}$  $+x^{2}$  $x$  
$12x^{2}$  $4x$  $+4$  $6x^{2}$  $+2x$  $2$  
$12x^{2}$  $4x$  $+4$  $2$  
$0$ 
Example 2. $\sqrt{x^{6}\!\!2x^{5}\!\!x^{4}\!+\!3x^{2}\!+\!2x\!+\!1}=$ ?
$\sqrt{}$  $(1$  $+2x$  $+3x^{2}$  $x^{4}$  $2x^{5}$  $+x^{6})$  $=$  $\pm$  $(1$  $+x$  $+x^{2}$  $x^{3})$  
$1$  $1$  
$2x$  $+3x^{2}$  $2$  $+x$  
$2x$  $+x^{2}$  $+x$  
$2x^{2}$  $x^{4}$  $2$  $+2x$  $+x^{2}$  
$2x^{2}$  $+2x^{3}$  $+x^{4}$  $x^{2}$  
$2x^{3}$  $2x^{4}$  $2x^{5}$  $+x^{6}$  $2$  $+2x$  $+2x^{2}$  $x^{3}$  
$2x^{3}$  $2x^{4}$  $2x^{5}$  $+x^{6}$  $x^{3}$  
$0$ 
Remark. The procedure may give a Taylor series expansion of the square root, if it is not a polynomial. E.g. we get
$\sqrt{1+x}=1+\frac{1}{2}x\frac{1}{8}x^{2}+\frac{1}{16}x^{3}\frac{5}{128}x^{4% }+...$ 
References
 1 Meyers Rechenduden. Erster verbesserter Neudruck. Bibliographisches Institut AG, Mannheim (1960).
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