Question 2: Find the square root of the following polynomials by division method:
\((iii)\) \(\;\) \(x^4-10x^3y+27x^2y^2-10xy^3+y^4 \)
Solution:
$\small
\begin{array}{ r l }
& \ \ \ \ \ \underline{\ \ \ \ \ \ x^{2} -5xy+y^{2} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
& x^{2}\bigr) \ \ \cancel{x^{4}} -10x^{3} y+27x^{2} y^{2} -10xy^{3} +y^{4}\\
& \ \ \ \ \ \ \underline{\underset{-}{+}\cancel{x^{4}} \ \ \ \ \ \ \ \ \ \ }\\
& 2x^{2} -5xy\ \overline{) -\cancel{10x^{3} y} +27x^{2} y^{2} -10xy^{3} +y^{4}}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\underset{+}{-} \ \cancel{10x^{3} y} \ \underset{-}{+} \ 25x^{2} y^{2} \ \ \ \ } \ \\
&2x^{2} -10xy+y^{2} \ \overline{)\ \ \ 2x^{2} y^{2} -10xy^{3} +y^{4}}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\underset{-}{+} 2x^{2} y^{2} \ \underset{-}{+} 10xy^{3} \ \underset{-}{+} y^{4} \ }\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\end{array}$
So, $\displaystyle \text{Square Root } =\pm \left( x^{2} -5xy+y^{2}\right)$