Question 2: Find HCF of the following expressions by using division method:
\((iii)\) \(\; \) \(2x^3+2x^2+2x+2,6x^3+12x^2+6x+12 \)
Solution:
$\begin{array}{ r l }
& \ \underline{\ \ \ \ \ \ \ \ 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
\small{2x^{3} +2x^{2} +2x+2} & \bigr) 6x^{3} +12x^{2} +6x+12\\
& \underline{\underset{\ \ -}{} 6x^{3} \underset{-}{+} 6x^{2} \ \underset{-}{+} 6x\underset{-}{+} 6}\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ 6x^{2} \ \ \ \ \ \ \ \ \ \ +6\\
\end{array}$
Here, $\displaystyle 6x^{2} +6=6\left( x^{2} +1\right) =2\times 3\left( x^{2} +1\right)$
$\displaystyle 3$ is not common in both given polynomials, so we ignore it and conisder only $\displaystyle 2x^{2} +2$
$\begin{array}{ r l }
& \ \underline{\ \ \ \ \ \ \ \ x+1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
2x^{2} +2 & \bigr) \ 2x^{3} +2x^{2} +2x+2\\
& \underline{\underset{\ \ -}{} 2x^{3} \ \ \ \ \ \ \ \ \ \ \ \underset{-}{+} 2x\ \ \ \ \ \ \ \ \ }\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2x^{2} \ \ \ \ \ \ \ \ +2\\
& \underline{\ \ \ \ \ \ \ \ \ \underset{-}{+} \ 2x^{2} \ \ \ \ \ \ \ \ \ \underset{-}{+\ } 2\ }\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\\
\end{array}$
$\displaystyle \text{ HCF} =2x^2+2$