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Question Number 175959 by BaliramKumar last updated on 10/Sep/22

$$f\left(x\right)=x^6-100x^5+100x^4-100x^3+100x^2-100x+100$$$$f\left(99\right)=?$$

Commented by infinityaction last updated on 10/Sep/22

$$\mathbf{p}={\mathbf{x}}^6+100\{(1-\mathbf{x})+{\mathbf{x}}^2(1-\mathbf{x})+{\mathbf{x}}^4(1-\mathbf{x})\}$$$$\mathbf{p}={\mathbf{x}}^6+100(1-\mathbf{x})\{1+{\mathbf{x}}^2+{\mathbf{x}}^4\}$$$$\mathit{G}.\mathit{P}.\mathbf{sum}\mathbf{of}1+{\mathbf{x}}^2+{\mathbf{x}}^4=\frac{{\mathbf{x}}^6-1}{{\mathbf{x}}^2-1}$$$$\mathbf{p}={\mathbf{x}}^6-\cancel{100(\mathbf{x}-1)}\times \frac{{\mathbf{x}}^6-1}{\cancel{(1+\mathbf{x})(\mathbf{x}-1)}}$$$$\mathbf{p}={\mathit{x}}^6-{\mathit{x}}^6+1=1$$

Commented by BaliramKumar last updated on 10/Sep/22

$$thankssir$$

Answered by Rasheed.Sindhi last updated on 10/Sep/22

$$f\left(x\right)=x^6-100x^5+100x^4-100x^3+100x^2-100x+100$$$$=x^5(x-100)+100x^4-100x^3+100x^2-100x+100$$$$f\left(99\right)=x^5(99-100)+100x^4-100x^3+100x^2-100x+100$$$$=-x^5+100x^4-100x^3+100x^2-100x+100$$$$=x^4(-x+100)-100x^3+100x^2-100x+100$$$$=x^4(-99+100)-100x^3+100x^2-100x+100$$$$=x^4-100x^3+100x^2-100x+100$$$$=x^3(x-100)+100x^2-100x+100$$$$=x^3(99-100)+100x^2-100x+100$$$$=-x^3+100x^2-100x+100$$$$=x^2(-x+100)-100x+100$$$$=x^2(-99+100)-100x+100$$$$=x^2-100x+100$$$$=x(x-100)+100$$$$=x(99-100)+100$$$$=-x+100$$$$=-99+100$$$$=1$$

Commented by BaliramKumar last updated on 10/Sep/22

$$\mathrm{thanks}\mathrm{sir}$$

Answered by Rasheed.Sindhi last updated on 10/Sep/22

$$f\left(99\right)=Remainderoff\left(x\right)/(x-99)$$$$\mathcal{B}y\mathcal{S}ynthetic\mathcal{D}ivision:$$$$\overline{)\begin{array}{cccccccc}99)& 1& -100& 100& -100& 100& -100& 100\\ & & +99& -99& +99& -99& 99& -99\\ & 1& -1& 1& -1& 1& -1& {\overline{)\begin{array}{c}1\end{array}}}^{}\end{array}}$$$$f\left(99\right)=1$$

Commented by Tawa11 last updated on 15/Sep/22

$$\mathrm{Great}\mathrm{sirs}.$$

Answered by Devendra291999 last updated on 10/Sep/22

$$\mathrm{Solution}$$$$\mathrm{Given}$$$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^6-{100\mathrm{x}}^5+{100\mathrm{x}}^4-{100\mathrm{x}}^3+{100\mathrm{x}}^2-100\mathrm{x}+100$$$$\mathrm{f}\left(99\right)=?$$$$\mathrm{then}$$$$\mathrm{x}=99\Rightarrow \mathrm{x}+1=100\mathrm{putt}$$$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^6-(\mathrm{x}+1){\mathrm{x}}^5+(\mathrm{x}+1){\mathrm{x}}^4-(\mathrm{x}+1){\mathrm{x}}^3+(\mathrm{x}+1){\mathrm{x}}^2-(\mathrm{x}+1)\mathrm{x}+\mathrm{x}+1$$$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{x}}^6-{\mathrm{x}}^6-{\mathrm{x}}^5+{\mathrm{x}}^5+{\mathrm{x}}^4-{\mathrm{x}}^3-{\mathrm{x}}^2+{\mathrm{x}}^3+{\mathrm{x}}^2-{\mathrm{x}}^2-\mathrm{x}+\mathrm{x}+1$$$$\mathrm{f}\left(\mathrm{x}\right)=1$$$$\mathrm{f}\left(99\right)=1\mathrm{Answer}$$

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