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Question Number 154959 by mathdanisur last updated on 23/Sep/21

$$\mathrm{Consider}\mathrm{the}\mathrm{polynomial}$$$$\mathrm{P}\left(\mathrm{x}\right)={\mathrm{x}}^{10}-{6\mathrm{x}}^9-{11\mathrm{x}}^8+{3\mathrm{x}}^2-18\mathrm{x}-7$$$$\mathrm{Calculate}:\mathrm{P}(3+2\sqrt{5})$$

Commented by prakash jain last updated on 23/Sep/21

$$x^{10}-6x^9-11x^8+3x^2-18x-7$$$$=x^8(x^2-6x-11)+3x^2-18x-7$$$$=x^8[{(x-3)}^2-20]+3x^2-18x+27-34$$$$=x^8\left(0\right)+3{(x-3)}^2-34$$$$=3\times 20-34$$$$=26$$

Commented by mathdanisur last updated on 23/Sep/21

$$\mathrm{Thankyou}\mathbf{S}\mathrm{er},\mathrm{very}\mathrm{nice}$$

Answered by Rasheed.Sindhi last updated on 23/Sep/21

$$\mathrm{P}\left(\mathrm{x}\right)={\mathrm{x}}^{10}-{6\mathrm{x}}^9-{11\mathrm{x}}^8+{3\mathrm{x}}^2-18\mathrm{x}-7;\mathrm{P}(3+2\sqrt{5})=?$$$$={\mathrm{x}}^{10}-{6\mathrm{x}}^9-{11\mathrm{x}}^8+{3\mathrm{x}}^2-18\mathrm{x}-33+26$$$$={\mathrm{x}}^8({\mathrm{x}}^2-6\mathrm{x}-11)+3({\mathrm{x}}^2-6\mathrm{x}-11)+26$$$$=\left(\underset{{\mathrm{P}}_1\left(\mathrm{x}\right)}{\underbrace{{\mathrm{x}}^2-6\mathrm{x}-11}}\right)({\mathrm{x}}^8+3)+26$$$$\mathrm{P}\left(\mathrm{x}\right)={\mathrm{P}}_1\left(\mathrm{x}\right).({\mathrm{x}}^8+3)+26$$$${\mathrm{P}}_1\left(\mathrm{x}\right)={(3+2\sqrt{5})}^2-6(3+2\sqrt{5})-11$$$$=9+20+12\sqrt{5}-18-12\sqrt{5}-11=0$$$$\mathrm{P}(3+2\sqrt{5})=\left(0\right)({(3+2\sqrt{5})}^8+3)+26=26$$$$\mathrm{P}(3+2\sqrt{5})=26$$$$$$

Commented by mathdanisur last updated on 23/Sep/21

$$\mathrm{Very}\mathrm{nice}\mathrm{solution}\mathbf{S}\mathrm{er},\mathrm{thankyou}$$

Answered by Rasheed.Sindhi last updated on 24/Sep/21

$$\mathrm{P}\left(\mathrm{x}\right)={\mathrm{x}}^{10}-{6\mathrm{x}}^9-{11\mathrm{x}}^8+{3\mathrm{x}}^2-18\mathrm{x}-7$$$$\underset{-}{\mathrm{P}(3+2\sqrt{5})=?}$$$$\mathbb{BY}\mathcal{S}ynthetic\mathcal{D}ivision$$$$\bullet If\mathrm{P}\left(\mathrm{x}\right)isdividedby\mathrm{x}-\mathrm{k}then$$$$remainder=\mathrm{P}\left(\mathrm{k}\right)$$$$\bullet \mathrm{k}iscalledmultiplyerandhere$$$$\mathrm{k}=3+2\sqrt{5}$$$$$$$$\left[\begin{array}{ccc}\underset{⌣}{3+2\sqrt{5}}& & \\ 1& & 1\\ -6& 3+2\sqrt{5}& -3+2\sqrt{5}\\ -11& 11& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 3& 0& 3\\ -18& 9+6\sqrt{5}& -9+6\sqrt{5}\\ -7& 33& 26\end{array}\right]$$$$\therefore \mathrm{P}(3+2\sqrt{5})=26$$

Commented by mathdanisur last updated on 24/Sep/21

$$\mathrm{Yes}\mathbf{S}\mathrm{er},\mathrm{cool},\mathrm{thank}\mathrm{you}$$

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