Kev kawm ntawm polynomial ntawm qib thib ob tau txais kev saib xyuas ntau hauv chav kawm qib yim algebra. Yog tias cov ntaub ntawv no tsis zoo los ntawm cov tub ntxhais kawm, cov teeb meem yuav tsum tsis txhob cuam tshuam ntawm kev xeem ntawm OGE thiab Kev Tshawb Fawb Hauv Xeev, ob qho tib si ntawm qib profile thiab hauv paus. Cov kev txawj uas yuav tsum tau muaj ntsig txog kev ua haujlwm plaub muaj xws li kev npaj thiab txheeb xyuas cov duab, daws qhov sib npaug.
Qhov kev faib ua feem ntawm square trinomial yog ib qho teeb meem hauv tsev kawm ntawv. Nws yog pab hauv kev daws qhov tsis sib xws los ntawm txoj kev ncua sij hawm.
Nrhiav cov hauv paus ntawm kev sib npaug
Thawj yam uas yuav tsum tau ua kom muaj ntau lub npe yog nrhiav nws cov hauv paus hniav.
Cov hauv paus yog cov lej uas tig cov lej ntawm cov monomial hauv cov lej rau xoom, uas zoo li cov duab sib tshuam nrog cov kab rov tav. Lawv txiav txim siab siv cov kev ntxub ntxaug lossis Vieta txoj kev xav.
Txoj kev sib cais ntawm trinomial ax2 + bx + c yog xam los ntawm tus qauv: D=b 2m- 4ac.
Thaum tus neeg ntxub ntxaug tsis zoo,cov hauv paus hniav yog qhia los ntawm nws thiab cov polynomial coefficients:
x1 =1/2(-b + √D); x2 =1/2(-b - √D)
Yog tus neeg ntxub ntxaug yog xoom, x1thiab x2yog tib yam.
Yuav daws qee qhov trinomials, nws yooj yim siv Vieta theorem:
x1 + x2 =-b: a; x1 × x2=c: a
Nws yuav siv qee qhov kev nkag siab ntawm lej los siv lub theorem. Cov kab hauv qab yog tias, paub cov lej thiab cov khoom ntawm ob yam tsis paub, khaws cov lej no. Yog tias lawv muaj, lawv yuav pom tshwj xeeb (mus txog qhov kev tso cai).
Koj tuaj yeem txheeb xyuas qhov siv tau ntawm lub theorem los ntawm kev suav cov lej thiab cov khoom ntawm cov hauv paus hniav hauv cov ntsiab lus dav dav. Cov qauv rau x1 thiab x2 kuj raug tshuaj xyuas los ntawm kev hloov pauv ncaj qha.
Factoring rule
Qhov teeb meem tuaj yeem daws tau los ntawm cov lej tiag yog tias tus lej muaj cov hauv paus hniav. Lub decomposition yog txiav txim los ntawm tus qauv:
ax2 + bx + c=a(x - x1)(x - x2 )
Examples
Teeb meem: nrhiav qhov factorization ntawm square trinomials.
a) x2 - 6x + 5
Kev daws: sau cov coefficients ntawm trinomial:
a=1; b=-6; c=5.
Siv Vieta theorem:
x1 + x2 =6;
x1 × x2=5.
Yuav pom tau tias x1 =1, x2 =5.
Yog tias, raws li kev sau ntawv sib npaug ntawm qhov theorem,Nws yog qhov ua tau kom pom cov hauv paus hniav sai sai, koj yuav tsum tam sim ntawd mus rau qhov kev suav ntawm cov neeg sib cais.
Tom qab pom cov hauv paus hniav, koj yuav tsum tau hloov lawv mus rau hauv cov qauv nthuav dav:
x2 - 6x + 5=(x - 1)(x - 5)
Qhov tshwm sim hauv daim foos no tuaj yeem suav tias yog qhov kawg.
b) 2x2 + x - 1
Kev daws:
a=2, b=1, c=-1.
Yog tias tus thawj coefficient txawv ntawm 1, kev siv Vieta theorem feem ntau yuav siv sij hawm ntau dua li kev daws los ntawm kev ntxub ntxaug, yog li cia peb mus rau kev suav nws.
D=1 - 4 × 2 × (-1)=9.
x1=1/2; x2=-1.
Tus qauv yog:
2x2 + x - 1=2(x - 1/2)(x + 1).
c)x2 - 8x + 16
Kev daws:
a=1; b=-8; c=16.
D=0.
Vim qhov kev ntxub ntxaug yog xoom, peb muaj qhov xwm txheej ntawm cov hauv paus hniav:
x1 =x2 =4.
qhov xwm txheej no, txawm li cas los xij, tsis yog qhov txawv ntawm qhov xav tau ua ntej.
x2 - 8x + 16=1(x - 4)(x - 4)
Qhov tshwm sim feem ntau sau li: (x - 4)2.
d)x2 - 7x + 1
Kev daws:
a=1; b=-7; c=1.
D=45.
Qhov piv txwv no txawv ntawm cov yav dhau los uas cov hauv paus ntsiab lus tsis tuaj yeem muab rho tawm los ntawm cov neeg ntxub ntxaug. Qhov no txhais tau hais tias cov hauv paus hniav ntawm polynomial yog irrational.
x1 =-1/2(7 + √45); x2 =-1/2(7 - √45).
Los yog sib npaug, x1=-3, 5 - 1/2√45; x2 =-3, 5 + 1/2√45.
Qhov kev xaiv kawg yog yooj yim dua los siv rau kev sau ntawv nthuav dav. Omitting tus coefficient laus, uas yog sib npaug ntawm no rau 1, peb tau txais:
x2- 7x + 1=(x + 3.5 + 1/2√45)(x + 3.5 - 1/2√45)
Rau rooj plaub thaum tus neeg ntxub ntxaug tsis zoo, cov lus teb hauv qab no yog txaus nyob rau hauv lub moj khaum ntawm lub tsev kawm ntawv cov ntaub ntawv kawm: lub trinomial tsis muaj cov hauv paus hniav thiab, yog li ntawd, tsis tuaj yeem factorized. Xws li trinomials kuj hu ua irreducible. Nws yog ib qho tseem ceeb kom nkag siab tias peb tab tom tham txog qhov muaj lossis tsis muaj keeb kwm tiag tiag.
Yog hais tias daim teb ntawm cov lej nyuaj raug txiav txim siab, qhov kev faib ua feem ntawm lub square trinomial yog ua tau nrog cov neeg sib cais.
Yam yuam kev
1) Thaum pib ntawm kev kawm ib tug polynomial, ntau tus neeg sau tawm cov coefficients tsis raug, piv txwv li, lawv tau xyuam xim rau qhov kev txiav txim ntawm cov monomials hauv cov ntawv sau.
Yog li qhov tseem ceeb ntawm qhov sib npaug 101 yog 79x + 38x2is 38, tsis yog 101 raws li koj xav.
Lwm qhov yuam kev cuam tshuam nrog cov coefficients ntawm qhov sib npaug yog qhov hu ua "poob ntawm kos npe". Hauv tib qhov piv txwv, coefficient b=-79, tsis yog 79.
2) Tau txais kev siv Vieta theorem rau rooj plaub thaum a=1, cov menyuam kawm ntawv qee zaum tsis nco qab txog nws cov qauv tsim. Nyob rau hauv lub polynomial los ntawm thawj kab lus, nws yog tsis raug rau xav tias cov sum ntawm cov hauv paus hniav yog 79, txij li thaum thawj coefficient txawv ntawm 1.
3) Kev suav yuam kev yog qhov teeb meem feem ntau rau cov tub ntxhais kawm. Feem ntau, kev kuaj xyuas pab kom tsis txhob muaj lawv.hloov.
Polynomials ntawm peb degree thiab siab dua
Polynomials ntawm qib siab dua tsis tshua pom hauv tsev kawm, vim qhov teeb meem ntawm kev nrhiav cov hauv paus rau polynomials ntawm qib peb thiab siab dua yog qhov nyuaj heev. Muaj algorithms ntawm high computational complexity rau expanding ib tug polynomial ntawm peb thiab plaub degree. Rau qib tsib thiab saum toj no, lub theorem ntawm unsolvability ntawm cov kab zauv nyob rau hauv radicals nyob rau hauv daim ntawv dav dav yog pov thawj.
Cov xwm txheej tshwj xeeb ntawm cov npe no, uas tuaj yeem txiav txim siab hauv tsev kawm theem siab, yog qhov muaj txiaj ntsig ntawm cov hauv paus ntsiab lus yooj yim xaiv. Tus lej ntawm qhov kawg tsis tuaj yeem tshaj qhov degree ntawm polynomial. Thaum ua haujlwm nrog lub dav hlau complex, lawv tus lej yog tib yam li qhov siab tshaj plaws.
Polynomials ntawm khib degree ib txwm muaj tsawg kawg yog ib lub hauv paus tiag. Qhov no yog ib qho yooj yim los qhia graphically - ib tug tas mus li muaj nuj nqi muab los ntawm xws li ib tug polynomial muaj ob qho tib si zoo thiab tsis zoo qhov tseem ceeb, uas txhais tau hais tias nws dhau los ntawm 0.
Tag nrho cov hauv paus hniav ntawm ob lub polynomials coincide yog thiab tsuas yog tias lawv cov coefficients yog proportional.
Feem ntau, qhov teeb meem ntawm kev nrhiav cov hauv paus hniav thiab qhov teeb meem ntawm kev tsim lub decomposition yuav suav hais tias sib npaug.
