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Category: Algebra

y-x-5-ax-4-bx-3-cx-2-dx-e-If-we-let-x-t-h-can-we-find-h-in-terms-of-a-b-c-d-e-such-that-y-t-R-t-2-pt-q-t-2-s-this-means-two-roots-are-of-opposite-sign-of-course-its-possible-by-shifting-the

Question Number 67819 by ajfour last updated on 31/Aug/19 $${y}={x}^{\mathrm{5}} +{ax}^{\mathrm{4}} +{bx}^{\mathrm{3}} +{cx}^{\mathrm{2}} +{dx}+{e} \\ $$$${If}\:{we}\:{let}\:{x}={t}+{h} \\ $$$${can}\:{we}\:{find}\:{h}\:{in}\:{terms}\:{of}\:{a},{b},{c},{d},{e} \\ $$$${such}\:{that} \\ $$$${y}=\left({t}+{R}\right)\left({t}^{\mathrm{2}} +{pt}+{q}\right)\left({t}^{\mathrm{2}} +{s}\right) \\…

Sum-to-n-terms-1-1-1-1-2-1-1-2-3-

Question Number 2273 by RasheedAhmad last updated on 12/Nov/15 $${Sum}\:{to}\:{n}\:{terms}. \\ $$$$\frac{\mathrm{1}}{\mathrm{1}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{1}+\mathrm{2}+\mathrm{3}}… \\ $$ Answered by Yozzi last updated on 12/Nov/15 $${For}\:{an}\:{A}.{P}\:{with}\:{first}\:{term}\:{a}\:{and} \\ $$$${common}\:{difference}\:{d},\:{its}\:{sum}\:{to}\:{n} \\…

a-Are-there-any-graphs-with-5-vertices-which-have-vertices-of-degrees-1-2-3-4-and-5-

Question Number 133318 by bramlexs22 last updated on 21/Feb/21 $$\left(\mathrm{a}\right)\:\mathrm{Are}\:\mathrm{there}\:\mathrm{any}\:\mathrm{graphs}\:\mathrm{with}\:\mathrm{5} \\ $$$$\mathrm{vertices}\:\mathrm{which}\:\mathrm{have}\:\mathrm{vertices}\:\mathrm{of}\: \\ $$$$\mathrm{degrees}\:\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}\:\mathrm{and}\:\mathrm{5}?\: \\ $$ Answered by EDWIN88 last updated on 21/Feb/21 $$\mathrm{No};\:\mathrm{if}\:\mathrm{a}\:\mathrm{graph}\:\mathrm{has}\:\mathrm{n}\:\mathrm{vertices}\:\mathrm{it}\:\mathrm{can}\:\mathrm{have}\:\mathrm{no}\: \\…

GENERALIZE-a-b-a-2-b-2-ab-a-3-b-3-a-b-c-a-2-b-2-c-2-ab-bc-ca-a-3-b-3-c-3-3abc-a-b-c-d-a-2-b-2-c-2-d-2-

Question Number 2240 by Rasheed Soomro last updated on 10/Nov/15 $$\mathcal{GENERALIZE}: \\ $$$$\left({a}+{b}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{ab}\right)={a}^{\mathrm{3}} +{b}^{\mathrm{3}} \\ $$$$\left({a}+{b}+{c}\right)\left({a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −{ab}−{bc}−{ca}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:={a}^{\mathrm{3}} +{b}^{\mathrm{3}} +{c}^{\mathrm{3}}…

hi-everybody-with-n-N-prove-that-n-0-N-n-n-0-n-2-2-n-

Question Number 133279 by greg_ed last updated on 20/Feb/21 $$\boldsymbol{\mathrm{hi}},\:\boldsymbol{\mathrm{everybody}}\:! \\ $$$$\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{n}}\:\in\:\mathbb{N}, \\ $$$$\boldsymbol{\mathrm{prove}}\:\boldsymbol{\mathrm{that}}\::\:\exists\:\boldsymbol{\mathrm{n}}_{\mathrm{0}} \:\in\:\mathbb{N}\:/\:\forall\:\boldsymbol{\mathrm{n}}\:\geqslant\:\boldsymbol{\mathrm{n}}_{\mathrm{0}} \:,\:\boldsymbol{\mathrm{n}}^{\mathrm{2}} \:\leqslant\:\mathrm{2}^{\boldsymbol{\mathrm{n}}} . \\ $$ Terms of Service Privacy Policy…

Find-the-value-of-1-cos-2-10-1-sin-2-20-1-sin-2-40-

Question Number 67711 by naka3546 last updated on 30/Aug/19 $${Find}\:\:{the}\:\:{value}\:\:{of} \\ $$$$\:\:\:\:\:\:\frac{\mathrm{1}}{\mathrm{cos}^{\mathrm{2}} \left(\mathrm{10}°\right)}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{20}°\right)}\:+\:\frac{\mathrm{1}}{\mathrm{sin}^{\mathrm{2}} \left(\mathrm{40}°\right)}\: \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com

Evaluate-1-2-3-1-3-3-2-4-3-3-by-considering-the-series-expansion-of-an-expression-of-the-form-P-x-e-x-where-P-x-is-a-suitably-chosen-polynomial-in-x-

Question Number 2157 by Yozzi last updated on 05/Nov/15 $${Evaluate}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{1}+\frac{\mathrm{2}^{\mathrm{3}} }{\mathrm{1}!}+\frac{\mathrm{3}^{\mathrm{3}} }{\mathrm{2}!}+\frac{\mathrm{4}^{\mathrm{3}} }{\mathrm{3}!}+… \\ $$$${by}\:{considering}\:{the}\:{series}\:{expansion} \\ $$$${of}\:{an}\:{expression}\:{of}\:{the}\:{form}\:{P}\left({x}\right){e}^{{x}} \\ $$$${where}\:{P}\left({x}\right)\:{is}\:{a}\:{suitably}\:{chosen} \\ $$$${polynomial}\:{in}\:{x}.\: \\ $$$$…