Question

Find the HCF of the following polynomials 12(x³+x²+x+1) and 18(x⁴ -1)

Answer 1

$$\begin{gathered} \mathrm{Let}\mathrm{P}=12\left[{\mathrm{x}}^3+{\mathrm{x}}^2+\mathrm{x}+1\right] \\ =12\left[{\mathrm{x}}^2\left(\mathrm{x}+1\right)+1\left(\mathrm{x}+1\right)\right] \\ =12\left({\mathrm{x}}^2+1\right)\left(\mathrm{x}+1\right) \\ =2\times 2\times 3\times \left({\mathrm{x}}^2+1\right)\left(\mathrm{x}+1\right) \\ \mathrm{Let}\mathrm{Q}=18\left[{\mathrm{x}}^4-1\right] \\ =18\left[{\left({\mathrm{x}}^2\right)}^2-1^2\right] \\ =18\left({\mathrm{x}}^2-1\right)\left({\mathrm{x}}^2+1\right) \\ =18\left(\mathrm{x}+1\right)\left(\mathrm{x}-1\right)\left({\mathrm{x}}^2+1\right) \\ =2\times 3\times 3\times \left(\mathrm{x}+1\right)\left(\mathrm{x}-1\right)\left({\mathrm{x}}^2+1\right) \\ \mathrm{HCF}=2\times 3\times \left({\mathrm{x}}^2+1\right)\left(\mathrm{x}+1\right)=6\left({\mathrm{x}}^2+1\right)\left(\mathrm{x}+1\right) \end{gathered}$$