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Question Number 184706 by Ml last updated on 10/Jan/23

$$\mathrm{deg}[3\mathrm{p}\left(\mathrm{x}\right)+\mathrm{Q}\left(\mathrm{x}\right)]=6$$$$\mathrm{deg}[\mathrm{p}\left(\mathrm{x}\right)+{\mathrm{x}}^4]=5$$$$\mathrm{deg}\left[\frac{({\mathrm{x}}^4+1)\mathrm{p}\left({\mathrm{x}}^2\right)}{{\mathrm{x}}^3\cdot \mathrm{Q}\left(\mathrm{x}\right)}\right]=?$$$$\mathrm{deg}=\mathrm{degree}$$

Answered by Rasheed.Sindhi last updated on 11/Jan/23

$$\mathrm{deg}[3\mathrm{p}\left(\mathrm{x}\right)+\mathrm{Q}\left(\mathrm{x}\right)]=6;\mathrm{deg}[\mathrm{p}\left(\mathrm{x}\right)+{\mathrm{x}}^4]=5$$$$\mathrm{deg}\left[\frac{({\mathrm{x}}^4+1)\mathrm{p}\left({\mathrm{x}}^2\right)}{{\mathrm{x}}^3\cdot \mathrm{Q}\left(\mathrm{x}\right)}\right]=?$$$$$$$$\mathrm{deg}[\mathrm{p}\left(\mathrm{x}\right)+{\mathrm{x}}^4]=5\Rightarrow \mathrm{deg}\left[\mathrm{p}\left(\mathrm{x}\right)\right]=5$$$$\mathrm{deg}[3\mathrm{p}\left(\mathrm{x}\right)+\mathrm{Q}\left(\mathrm{x}\right)]=6\Rightarrow \mathrm{deg}\left[\mathrm{Q}\left(\mathrm{x}\right)\right]=6$$$$\mathrm{p}\left({\mathrm{x}}^2\right)=10\Rightarrow \mathrm{deg}\left[({\mathrm{x}}^4+1)\mathrm{p}\left({\mathrm{x}}^2\right)\right]=14$$$$\mathrm{deg}[{\mathrm{x}}^3\cdot \mathrm{Q}\left(\mathrm{x}\right)]=9$$$$\mathrm{deg}\left[\frac{({\mathrm{x}}^4+1)\mathrm{p}\left({\mathrm{x}}^2\right)}{{\mathrm{x}}^3\cdot \mathrm{Q}\left(\mathrm{x}\right)}\right]=14-9=5$$$$\mathrm{provided}\mathrm{that}{\mathrm{x}}^3\cdot \mathrm{Q}\left(\mathrm{x}\right)\mid ({\mathrm{x}}^4+1)\mathrm{p}\left({\mathrm{x}}^2\right)$$

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