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Category: Algebra

Question-133180

Question Number 133180 by mathlove last updated on 19/Feb/21 Commented by mr W last updated on 19/Feb/21 $${no}\:{real}\:{solution}! \\ $$$${x}^{{x}} \geqslant\frac{\mathrm{1}}{\:\sqrt[{{e}}]{{e}}}\approx\mathrm{0}.\mathrm{692} \\ $$$$\frac{\mathrm{1}}{\mathrm{256}}<<\mathrm{0}.\mathrm{692} \\ $$$$\Rightarrow{x}^{{x}}…

Solve-x-2-1-lt-5-

Question Number 67574 by pete last updated on 28/Aug/19 $$\mathrm{Solve}\:\mathrm{x}^{\mathrm{2}} +\mathrm{1}<−\mathrm{5} \\ $$ Commented by mathmax by abdo last updated on 28/Aug/19 $$\left({e}\right)\Rightarrow{x}^{\mathrm{2}} +\mathrm{6}<\mathrm{0}\:\:\:{impossible}\:{equation}\:\Rightarrow{no}\:{solution}\: \\…

Show-that-5-2-3-1-4-is-in-F-2-by-expressing-the-number-in-form-a-1-b-1-k-1-where-a-1-b-1-k-1-are-in-F-1-

Question Number 133072 by bemath last updated on 18/Feb/21 $$\mathrm{Show}\:\mathrm{that}\:\frac{\mathrm{5}}{\mathrm{2}−\sqrt[{\mathrm{4}}]{\mathrm{3}}}\:\mathrm{is}\:\mathrm{in}\:\mathrm{F}_{\mathrm{2}} \:\mathrm{by}\:\mathrm{expressing} \\ $$$$\mathrm{the}\:\mathrm{number}\:\mathrm{in}\:\mathrm{form}\:{a}_{\mathrm{1}} +{b}_{\mathrm{1}} \sqrt{{k}_{\mathrm{1}} }\:\mathrm{where} \\ $$$${a}_{\mathrm{1}} ,{b}_{\mathrm{1}} ,\:{k}_{\mathrm{1}} \:{are}\:{in}\:{F}_{\mathrm{1}} \\ $$ Answered by…

Find-a-non-constant-function-f-satisfying-f-0-1-f-2-0-and-f-x-y-f-x-f-y-f-2-x-f-y-2-

Question Number 2000 by Yozzi last updated on 29/Oct/15 $${Find}\:{a}\:{non}−{constant}\:{function}\:{f}\: \\ $$$${satisfying}\:{f}\left(\mathrm{0}\right)=\mathrm{1},{f}\left(−\mathrm{2}\right)=\mathrm{0}\:{and} \\ $$$${f}\left({x}−{y}\right)={f}\left({x}\right){f}\left({y}\right)−{f}\left(−\mathrm{2}−{x}\right){f}\left({y}−\mathrm{2}\right). \\ $$ Commented by prakash jain last updated on 29/Oct/15 $${x}=\mathrm{0}…

x-2-f-x-f-x-2-f-x-

Question Number 1988 by Rasheed Soomro last updated on 28/Oct/15 $${x}^{\mathrm{2}} =\:\frac{{f}\left({x}\right)+{f}\left(−{x}\right)}{\mathrm{2}} \\ $$$${f}\left({x}\right)=? \\ $$ Answered by 123456 last updated on 28/Oct/15 $$\mathrm{supossing}\:\mathrm{that}\:{f}\:\mathrm{is}\:\mathrm{poly} \\…

When-the-polynomial-f-x-is-divided-by-x-2-the-remainder-is-4-and-when-it-is-divided-x-3-the-remainder-is-7-Given-that-f-x-may-be-written-in-the-formf-x-x-2-x-3-Q-x-ax-b-find-the-remainder-

Question Number 133053 by pete last updated on 18/Feb/21 $$\mathrm{When}\:\mathrm{the}\:\mathrm{polynomial}\:\mathrm{f}\left({x}\right)\:\mathrm{is}\:\mathrm{divided}\:\mathrm{by} \\ $$$$\left(\mathrm{x}−\mathrm{2}\right)\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{4}\:\mathrm{and}\:\mathrm{when}\:\mathrm{it}\:\mathrm{is}\:\mathrm{divided} \\ $$$$\left(\mathrm{x}−\mathrm{3}\right)\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{is}\:\mathrm{7}.\:\mathrm{Given}\:\mathrm{that}\:\mathrm{f}\left({x}\right) \\ $$$$\mathrm{may}\:\mathrm{be}\:\mathrm{written}\:\mathrm{in}\:\mathrm{the}\:\mathrm{formf}\left({x}\right)=\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right)\mathrm{Q}\left(\mathrm{x}\right)+\mathrm{ax}+\mathrm{b}, \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{remainder}\:\mathrm{when}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{divided} \\ $$$$\mathrm{by}\:\left(\mathrm{x}−\mathrm{2}\right)\left(\mathrm{x}−\mathrm{3}\right).\:\mathrm{If}\:\mathrm{also}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{is}\:\mathrm{a}\:\mathrm{cubic}\:\mathrm{function} \\ $$$$\mathrm{in}\:\mathrm{which}\:\mathrm{the}\:\mathrm{coefficient}\:\mathrm{of}\:\mathrm{x}^{\mathrm{3}} \:\mathrm{is}\:\mathrm{unity}\:\mathrm{and} \\ $$$$\mathrm{f}\left(\mathrm{1}\right)=\mathrm{1},\:\mathrm{determine}\:\mathrm{Q}\left(\mathrm{x}\right).…

Show-that-1n-3-2n-3n-2-is-divisible-by-2-and-3-for-all-positive-integers-n-

Question Number 67501 by TawaTawa last updated on 28/Aug/19 $$\mathrm{Show}\:\mathrm{that}\:\:\mathrm{1n}^{\mathrm{3}} \:+\:\mathrm{2n}\:+\:\mathrm{3n}^{\mathrm{2}} \:\:\mathrm{is}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{2}\:\mathrm{and}\:\mathrm{3}\:\mathrm{for}\:\mathrm{all}\:\mathrm{positive}\:\mathrm{integers}\:\mathrm{n}. \\ $$ Commented by Prithwish sen last updated on 28/Aug/19 $$\mathrm{Another}\:\mathrm{approch} \\ $$$$\mathrm{We}\:\mathrm{know}\:\mathrm{that}…