Question 2: Find the square root of the following polynomials by division method:
\((i)\) \(\quad\) \( 4x^4-28x^3+37x^2+42x+9\)
Solution:
$\begin{array}{ r l }
& \ \ \ \ \ \ \ \underline{\ \ \ \ \ \ 2x^{2} -7x-3\ \ \ \ \ \ \ \ }\\
& 2x^{2}\bigr) \ \cancel{4x^{4}} -28x^{3} +37x^{2} +42x+9\\
&\ \ \ \ \ \ \underline{ \underset{-}{+} \ \cancel{4x^{4}} \ \ \ \ \ \ \ \ \ \ }\\
& 4x^{2} -7x\ \overline{) -\cancel{28x^{3}} +37x^{2} +42x+9}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\underset{+}{-} \ \cancel{28x^{3}} \ \underset{-}{+} \ 49x^{2} \ \ \ \ } \ \\
& 4x^{2} -14x-3\ \overline{) -12x^{2} +42x+9}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \underline{\underset{+}{-} \ 12x^{2} \ \underset{-}{+} 42x\ \underset{-}{+} \ 9\ }\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0
\end{array}$
So, $\displaystyle \text{Square Root } =\pm \left( 2x^{2} -7x-3\right)$