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Category: Algebra

If-P-x-aX-3-bX-c-Q-X-a-0-has-the-roots-x-1-x-2-and-x-3-such-that-x-1-x-2-x-3-then-prove-that-x-1-0-if-and-only-if-b-c-

Question Number 153161 by mathdanisur last updated on 05/Sep/21 $$\mathrm{If}\:\:\mathrm{P}\left(\mathrm{x}\right)=\mathrm{aX}^{\mathrm{3}} +\mathrm{bX}+{c}\:\in\:\mathrm{Q}\left[\mathrm{X}\right]\:\:;\:\:\left(\mathrm{a}\neq\mathrm{0}\right) \\ $$$$\mathrm{has}\:\mathrm{the}\:\mathrm{roots}\:\:\mathrm{x}_{\mathrm{1}} \:,\:\mathrm{x}_{\mathrm{2}} \:\:\mathrm{and}\:\:\mathrm{x}_{\mathrm{3}} \\ $$$$\mathrm{such}\:\mathrm{that}\:\:\mathrm{x}_{\mathrm{1}} \:=\:\mathrm{x}_{\mathrm{2}} \mathrm{x}_{\mathrm{3}} \\ $$$$\mathrm{then}\:\mathrm{prove}\:\mathrm{that}\:\mathrm{x}_{\mathrm{1}} \:=\:\mathrm{0}\:\mathrm{if}\:\mathrm{and}\:\mathrm{only}\:\mathrm{if}\:\mathrm{b}=\mathrm{c} \\ $$ Answered…

Find-the-set-x-x-b-a-b-a-2-b-2-a-gt-0-b-gt-0-

Question Number 153160 by mathdanisur last updated on 05/Sep/21 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{set}: \\ $$$$\left\{\:\mathrm{x}\:\mid\:\mathrm{x}\:=\:\frac{\mathrm{b}\centerdot\left(\mathrm{a}\:+\:\mathrm{b}\right)}{\mathrm{a}^{\mathrm{2}} \:+\:\mathrm{b}^{\mathrm{2}} }\:\:;\:\:\mathrm{a}>\mathrm{0}\:,\:\mathrm{b}>\mathrm{0}\right\} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Determine-all-the-derivable-functions-f-R-R-which-satisfy-f-x-3-f-0-f-x-f-0-x-2-x-0-

Question Number 153162 by mathdanisur last updated on 05/Sep/21 $$\mathrm{Determine}\:\mathrm{all}\:\mathrm{the}\:\mathrm{derivable}\:\mathrm{functions} \\ $$$$\mathrm{f}:\mathbb{R}\rightarrow\mathbb{R}\:\:\mathrm{which}\:\mathrm{satisfy} \\ $$$$\frac{\mathrm{f}\left(\mathrm{x}^{\mathrm{3}} \right)\:-\:\mathrm{f}\left(\mathrm{0}\right)}{\mathrm{f}\left(\mathrm{x}\right)\:-\:\mathrm{f}\left(\mathrm{0}\right)}\:=\:\mathrm{x}^{\mathrm{2}} \:\:;\:\:\forall\mathrm{x}\neq\mathrm{0} \\ $$ Answered by aleks041103 last updated on 05/Sep/21…

If-A-is-a-fifty-element-subset-of-the-set-1-2-3-100-such-that-no-two-numbers-from-A-add-up-to-100-show-that-A-contains-a-square-

Question Number 22082 by Tinkutara last updated on 10/Oct/17 $$\mathrm{If}\:{A}\:\mathrm{is}\:\mathrm{a}\:\mathrm{fifty}-\mathrm{element}\:\mathrm{subset}\:\mathrm{of}\:\mathrm{the}\:\mathrm{set} \\ $$$$\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:….,\:\mathrm{100}\right\}\:\mathrm{such}\:\mathrm{that}\:\mathrm{no}\:\mathrm{two} \\ $$$$\mathrm{numbers}\:\mathrm{from}\:{A}\:\mathrm{add}\:\mathrm{up}\:\mathrm{to}\:\mathrm{100}\:\mathrm{show} \\ $$$$\mathrm{that}\:{A}\:\mathrm{contains}\:\mathrm{a}\:\mathrm{square}. \\ $$ Answered by Rasheed.Sindhi last updated on 14/Oct/17…

if-x-y-2-xy-1-5-2-and-x-2-y-2-5-find-1-x-1-y-

Question Number 153128 by mathdanisur last updated on 04/Sep/21 $$\mathrm{if}\:\:\:\frac{\mathrm{x}+\mathrm{y}}{\mathrm{2}}\:+\:\sqrt{\mathrm{xy}}\:=\:\frac{\mathrm{1}+\sqrt{\mathrm{5}}}{\mathrm{2}}\:\:\mathrm{and}\:\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\sqrt{\mathrm{5}} \\ $$$$\mathrm{find}\:\:\:\frac{\mathrm{1}}{\mathrm{x}}\:+\:\frac{\mathrm{1}}{\mathrm{y}}\:=\:? \\ $$ Answered by EDWIN88 last updated on 05/Sep/21 $$\:\:\begin{cases}{{x}+{y}={u}}\\{{xy}={v}}\end{cases}\Rightarrow\begin{cases}{{u}+\mathrm{2}\sqrt{{v}}\:=\mathrm{1}+\sqrt{\mathrm{5}}}\\{{u}^{\mathrm{2}} −\mathrm{2}{v}=\sqrt{\mathrm{5}}}\end{cases}…