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Question Number 64508 by lalitchand last updated on 18/Jul/19

$$\mathrm{factorize}(\mathrm{x}+1)(\mathrm{x}+3)(\mathrm{x}+5)(\mathrm{x}+7)+16$$

Commented by Prithwish sen last updated on 18/Jul/19

$$(\mathrm{x}+1)(\mathrm{x}+7)(\mathrm{x}+3)(\mathrm{x}+5)+16$$$$=({\mathrm{x}}^2+8\mathrm{x}+7)({\mathrm{x}}^2+8\mathrm{x}+15)+16$$$$=(\mathrm{a}+7)(\mathrm{a}+15)+16\mathrm{putting}{\mathrm{x}}^2+8\mathrm{x}=\mathrm{a}$$$$={\mathrm{a}}^2+22\mathrm{a}+121$$$$={(\mathrm{a}+11)}^2$$$$=({\mathrm{x}}^2+8\mathrm{x}+11)({\mathrm{x}}^2+8\mathrm{x}+11)$$

Answered by behi83417@gmail.com last updated on 18/Jul/19

$$(\mathrm{x}+1)(\mathrm{x}+7)={\mathrm{x}}^2+8\mathrm{x}+7$$$$(\mathrm{x}+3)(\mathrm{x}+5)={\mathrm{x}}^2+8\mathrm{x}+15$$$${\mathrm{x}}^2+8\mathrm{x}=\mathrm{p}\Rightarrow (\mathrm{p}+7)(\mathrm{p}+15)+16=$$$$={\mathrm{p}}^2+22\mathrm{p}+121={(\mathrm{p}+11)}^2$$$$\Rightarrow (\mathrm{x}+1)(\mathrm{x}+3)(\mathrm{x}+5)(\mathrm{x}+7)+16=$$$$={({\mathbf{x}}^2+8\mathbf{x}+11)}^2.◼$$

Answered by MJS last updated on 18/Jul/19

$$\mathrm{put}x=t-4$$$$(t-3)(t-1)(t+1)(t+3)+16$$$$t^4-10t^2+25$$$${(t^2-5)}^2$$$$\mathrm{put}t=x+4$$$${({(x+4)}^2-5)}^2$$$${(x+4-\sqrt{5})}^2{(x+4+\sqrt{5})}^2$$$$\mathrm{or}$$$${(x^2+8x+11)}^2$$

Commented by Prithwish sen last updated on 18/Jul/19

$$\mathrm{great}\mathrm{idea}\mathrm{sir}.$$

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