Hauv tsab xov xwm no peb yuav sim ChatGPT txoj kev paub txog lej. Peb yuav sim coj kom zoo dua ntawm kev txawj ntse txawj ntse los nrhiav ib qho piv txwv rau Fundamental Theorem of Algebra, nrhiav pom tias nws yuav ua rau peb xav tsis thoob rau Fields Medal.
Yog tias peb nug txog cov hauv paus ntsiab lus ntawm polynomial ntawm degree 3, qhov no txhua yam tiag tiag, ChatGPT qhia peb tias qhov kev daws teeb meem kev tshuaj ntsuam xyuas yuav nyob ntawm qhov kev thov polynomial, yog li peb pom zoo kom siv cov lej lej xws li Newton-Raphson txoj kev.
Ib qho yuam kev nyob rau hauv kev xam ntawm cov derivative
Txog tam sim no, peb tsis tuaj yeem ua xyem xyav txog kev muaj peev xwm ua lej ntawm AI, yog li peb sim kom tau txais nws los daws qhov teeb meem ntawm kev nrhiav cov hauv paus hniav ntawm polynomial p(x) = x3 - 3 × 2 + 4 thiab ua rau peb xav tsis thoob nws ua qhov kev suav. tsis ncaj ncees lawm, qhov derivative, yog li tau txais cov hauv paus hniav tsis raug. Nws rov x = 0 raws li lub hauv paus ntawm lub polynomial thiab peb hais kom nws mus xyuas nws. Lawm, koj paub qhov muaj qhov yuam kev tab sis koj tsis paub qhov twg nws tshwm sim. Peb tau pom tias qhov yuam kev yog nyob rau hauv lub derivative ntawm lub polynomial thiab peb nug tias nws tau raug xam los ntawm cov hauv paus hniav los ntawm txoj kev Newton-Raphson. Kuj ceeb tias, nws ua qhov yuam kev rov ua dua lub sijhawm no hauv kev ua haujlwm yooj yim, raws li peb tuaj yeem pom hauv cov duab hauv qab no:
Kev suav yuam kev
Thaum pom qhov yuam kev hauv kev suav, peb nug lawv dua, ua lwm qhov yuam kev, yog li peb muab lawv thawj iteration ntawm Newton-Raphson Method, uas yog, x₁ = 5/3 thiab peb thov kom txuas ntxiv cov iterations, ua rau x₁ = 5/3 yog lub hauv paus ntawm polynomial. Peb corroborate los ntawm kev nug dua yog tias tus nqi 5/3 yog lub hauv paus ntawm polynomial, thiab peb tau txais cov lus teb zoo. Peb thov kom suav tus nqi ntawm cov polynomial hauv tus nqi ntawd, thiab, vim tias qhov tshwm sim txawv ntawm xoom, peb qhia meej tias nws tsis tuaj yeem yog lub hauv paus. Nws to taub nws thiab exonerates nws tus kheej raws li peb tau pom hauv qab no:
Peb xaus tias txoj kev xav ntawm Newton-Raphson Method yog qhov tseeb, tab sis nws daim ntawv thov tsis yog, yog li peb sim nrhiav cov hauv paus hniav siv lwm txoj kev, xws li polynomial factorization.
Hauv qhov no, peb pom tias cov hauv paus hniav ntawm polynomial p(x) yog x = r thiab x = 1 ± 2i.
Thaum nug kom paub tseeb tias tus nqi ntawm p (1 + 2i) txawv ntawm xoom thiab yog li tsis tuaj yeem yog lub hauv paus ntawm peb lub npe, nws rov lees paub qhov yuam kev. Thaum peb mus txog qhov xwm txheej no, peb mus nrog cov lus qhia, thiab peb qhia nws tias x = - 1 yog lub hauv paus tiag tiag ntawm cov lej thiab cov hauv paus ntsiab lus suav. Nws thawj cov lus teb tsis tuaj yeem xav tsis thoob, qhia peb tias ntxiv rau x = – 1, lwm cov hauv paus hniav ntawm polynomial p(x) = 4 – 3 × 2 + x3 yog x = 1 + 2i thiab x = 1 – 2i. . Nws muab cov txiaj ntsig tsis raug txog plaub zaug, yog li peb tsis muaj kev xaiv tab sis muab nws nrog lub hauv paus tshiab. Hauv qhov no, tsis yog muab rau koj, peb nug yog tias x = 2 yog lub hauv paus ntawm peb cov polynomial. Txiav txim rau koj tus kheej cov lus teb, lossis theej, cov kev suav uas ChatGPT ua kom paub tseeb tias x = 2 tsis yog lub hauv paus:
Tom qab kuaj xyuas cov kev suav dua, ua kom tiav los ntawm kev piav qhia tias cov hauv paus hniav ntawm peb cov polynomial yog x = - 1, x = 1 thiab x = 2.
Peb yuav qhia koj tias peb lub hauv paus yog, los ntawm kev xa rov qab qhov tseeb, uas yog, x = 1 tsis yog lub hauv paus thaum lwm qhov ob qhov tseem ceeb. Peb tsis tso tseg thiab sim mus nrhiav qhov thib peb hauv paus ntawm polynomial, thiab qhov ntawd yog qhov uas peb pom qhov yuam kev loj tshaj ntawm kev txhais lus. Peb cov lus piav qhia: “Lub Ntsiab Lus Tseem Ceeb ntawm Algebra tsim kom txhua tus lej ntawm kev kawm tiav muaj cov hauv paus ntsiab lus, ob qho tib si tiag tiag thiab nyuaj. Yog li, yog hais tias ib tug polynomial ntawm degree n muaj k tiag cov hauv paus hniav, ces nws yuav tsum muaj nk complex keeb kwm. " Txog rau qhov no peb pom zoo. Peb txuas ntxiv nrog: “Nyob rau hauv cov ntaub ntawv ntawm lub polynomial p(x) = 4 – 3 × 2 + x3, peb tau pom tias nws muaj ob lub hauv paus tiag tiag, x = – 1 thiab x = 2. Vim p(x) yog a. polynomial ntawm degree 3, ces nws yuav tsum muaj ib tug ntxiv complex cag. Lub hauv paus complex no yog x = 1 ± 2i. Peb tsis tuaj yeem hla peb txoj kev xav tsis thoob thiab peb xav tias nws tsuas xav qhia peb ib qho ntawm ob lub hauv paus, yog li peb muab sijhawm rau nws, ua rau:
Yog li yog tias peb hais yog lawm, peb nyuam qhuav pom qhov piv txwv rau Fundamental Theorem of Algebra, polynomial ntawm degree 3 nrog 4 cov hauv paus. Puas yog peb mus rau Fields Medal?
Lub AI tau rov hais dua tias nws cov lus teb yog qhov tseeb ntawm ob lub sijhawm ntxiv, uas qhia tias polynomial ntawm degree 3 tuaj yeem muaj 4 cov hauv paus hniav. Peb txawm hais kom nrhiav tau lawv siv Txoj Kev Bisection. Tam sim no, peb tso tseg txuas ntxiv mus nrhiav cov hauv paus hniav ntawm ib qho yooj yim polynomial ntawm degree 3. Peb hais goodbye cordially nrog ib ntsiav tshuaj kawg:
Raws li cov ntsiab lus kawg, peb tsis tau hais tias ChatGPT yog qhov tsis zoo Artificial Intelligence, nyob deb ntawm nws, tsuas yog qhov rov qab, nws yog AI zoo heev, tab sis hauv nws daim teb, hauv Natural Language Processing, txawm hais tias hauv lej nws tseem muaj ntau. ua. kawm. Peb yuav tsum ua tib zoo saib xyuas cov txiaj ntsig uas cov cav rov qab los rau peb: lawv tsis muaj tseeb txawm hais tias lawv piav qhia tau zoo npaum li cas, thiab zoo li tsis muaj tib neeg tuaj yeem txheeb xyuas lawv qhov tseeb.
QHOV TSEEB
Íñigo Sarría Martínez De Mendivil
Tsab ntawv xov xwm no tau luam tawm thawj zaug ntawm Kev Sib Tham.
