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Category: Algebra

Consider-a-polynomial-equation-i-0-n-a-i-x-i-0-a-i-Z-Prove-that-if-a-b-c-is-a-root-of-the-above-equation-then-a-b-c-is-also-a-root-a-b-c-Z-c-is-not-a-whole-square-

Question Number 3451 by prakash jain last updated on 13/Dec/15 $$\mathrm{Consider}\:\mathrm{a}\:\mathrm{polynomial}\:\mathrm{equation} \\ $$$$\underset{{i}=\mathrm{0}} {\overset{{n}} {\sum}}{a}_{{i}} {x}^{{i}} =\mathrm{0},\:{a}_{{i}} \in\mathbb{Z} \\ $$$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}\:{a}+{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{a}\:\mathrm{root}\:\mathrm{of}\:\mathrm{the}\:\mathrm{above} \\ $$$$\mathrm{equation}\:\mathrm{then}\:{a}−{b}\sqrt{{c}}\:\mathrm{is}\:\mathrm{also}\:\mathrm{a}\:\mathrm{root}. \\ $$$${a},{b},{c}\in\mathbb{Z},\:{c}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{whole}\:\mathrm{square}. \\…

determinant-If-x-2-2y-5-and-y-2-2x-5-what-the-value-of-x-3-2x-2-y-2-y-2-

Question Number 134467 by bramlexs22 last updated on 04/Mar/21 $$\begin{array}{|c|c|}{\mathrm{If}\:\mathrm{x}^{\mathrm{2}} =\mathrm{2y}+\mathrm{5}\:\mathrm{and}\:\mathrm{y}^{\mathrm{2}} =\mathrm{2x}+\mathrm{5}\:}\\{\mathrm{what}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} −\mathrm{2x}^{\mathrm{2}} \mathrm{y}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} .}\\\hline\end{array} \\ $$ Answered by benjo_mathlover last updated on 04/Mar/21…

Question-68912

Question Number 68912 by TawaTawa last updated on 16/Sep/19 Answered by MJS last updated on 17/Sep/19 $$\left(\mathrm{1}\right)\:\:{a}^{\mathrm{2}} +{ab}+{b}^{\mathrm{2}} −\mathrm{7}=\mathrm{0} \\ $$$$\left(\mathrm{2}\right)\:\:{b}^{\mathrm{2}} +{bc}+{c}^{\mathrm{2}} −\mathrm{21}=\mathrm{0} \\ $$$$\left(\mathrm{3}\right)\:\:{c}^{\mathrm{2}}…