Polynomial, lossis polynomial - ib qho ntawm cov qauv ntawm cov lej lej, uas muaj nyob hauv tsev kawm ntawv thiab lej siab dua. Txoj kev kawm ntawm ib tug polynomial yog lub ntsiab lus tseem ceeb tshaj plaws hauv chav kawm algebra, vim, ntawm ib sab, polynomials yog qhov yooj yim heev piv rau lwm hom haujlwm, thiab, ntawm qhov tod tes, lawv tau siv dav hauv kev daws teeb meem ntawm kev txheeb xyuas lej.. Yog li ntawd yog dab tsi yog ib tug polynomial?
Definition
Lub ntsiab lus ntawm lo lus polynomial tuaj yeem muab los ntawm lub tswvyim ntawm monomial, lossis monomial.
A monomial is an expression of form cx1i1x2 i2 …xin. Ntawm no yog qhov tsis tu ncua, x1, x2, … x - hloov pauv, i1, i2, … - exponents ntawm variables. Tom qab ntawd ib tug polynomial yog ib qho finite sum ntawm monomials.
Yuav kom nkag siab tias tus lej yog dab tsi, koj tuaj yeem saib cov piv txwv tshwj xeeb.
Lub square trinomial, sib tham nthuav dav hauv chav kawm lej 8th, yog ib tug polynomial: ax2+bx+c.
A polynomial nrog ob qhov sib txawv yuav zoo li no: x2-xy+y2. Xws liib tug polynomial tseem hu ua ib qho tsis tiav square ntawm qhov txawv ntawm x thiab y.
polynomial classifications
Polynomial degree
Rau txhua tus monomial nyob rau hauv lub polynomial, nrhiav cov lej ntawm cov exponents i1+i2+…+in. Qhov loj tshaj ntawm cov sums yog hu ua tus exponent ntawm lub polynomial, thiab cov monomial sib raug rau cov lej no yog hu ua lub sij hawm siab tshaj.
Los ntawm txoj kev, ib qho tas mus li tuaj yeem suav tias yog polynomial ntawm degree xoom.
Txo thiab tsis txo cov polynomials
Yog tus coefficient c yog sib npaug rau 1 rau lub sij hawm siab tshaj, ces tus polynomial muab, txwv tsis pub nws tsis yog.
Piv txwv li, cov lus x2 + 2x + 1 yog txo qis, thiab 2x 2+ 2x + 1 tsis txo.
Homogeneous thiab inhomogeneous polynomials
Yog cov qib ntawm txhua tus tswv cuab ntawm ib lub npe sib npaug, ces peb hais tias xws li ib tug polynomial yog homogeneous. Tag nrho lwm cov polynomials suav hais tias tsis yog homogeneous.
Homogeneous polynomials: x2-xy+y2, xyz+x3 +y 3. Heterogeneous: x+1, x2+y.
Muaj cov npe tshwj xeeb rau ntau lub npe ntawm ob thiab peb lo lus: binomial thiab trinomial, feem.
Polynomials ntawm ib qho kev sib txawv yog faib rau hauv ib pawg.
Daim ntawv thov ntawm ntau lub npe ntawm ib qho sib txawv
Polynomials ntawm ib qho kev sib txawv approximate zoo txuas ntxiv ua haujlwm ntawm kev sib txawv ntawm qhov sib txawv ntawm ib qho kev sib cav.
Qhov tseeb yog tias cov polynomials no tuaj yeem suav tias yog ib feem ntawm cov khoom siv hluav taws xob, thiab kev ua haujlwm tas mus li tuaj yeem sawv cev ua cov koob nrog qhov yuam kev me me. Lub expansion series ntawm ib tug muaj nuj nqi hu ua Taylor series, thiab lawvCov nyiaj ib nrab nyob rau hauv daim ntawv ntawm polynomials - Taylor polynomials.
Kawm graphically tus cwj pwm ntawm kev ua haujlwm los ntawm kwv yees nws nrog qee tus lej feem ntau yooj yim dua li kev tshawb nrhiav tib txoj haujlwm ncaj qha lossis siv cov koob.
Nws yog ib qho yooj yim mus nrhiav derivatives ntawm polynomials. Txhawm rau nrhiav cov hauv paus ntawm polynomials ntawm qib 4 thiab hauv qab no, muaj cov qauv npaj ua tiav, thiab rau kev ua haujlwm nrog qib siab dua, siv cov txheej txheem siab-precision approximate.
Tseem kuj tseem muaj qhov dav dav ntawm cov lus piav qhia rau kev ua haujlwm ntawm ntau qhov sib txawv.
Newton's binomial
Ntiaj teb polynomials yog Newton's polynomials, muab los ntawm cov kws tshawb fawb los nrhiav cov coefficients ntawm kev qhia (x + y).
Nws yog txaus los saib thawj ob peb lub zog ntawm binomial decomposition kom paub tseeb tias cov mis tsis tseem ceeb:
(x+y)2=x2+2xy+y2;
(x+y)3=x3+3x2y+3xy2+y3;
(x+y)4=x4+4x3y+6x2y2+4xy3+y4;
(x+y)5=x5+5x4y+10x3y2+10x2y3+5xy 4+y5.
Rau txhua tus coefficient muaj ib qho kev qhia uas tso cai rau koj los xam nws. Txawm li cas los xij, nco txog cov qauv tsim nyog thiab ua cov haujlwm lej lej tsim nyog txhua zaus yuav tsis yooj yim rau cov lej uas feem ntau xav tau kev nthuav dav. Pascal daim duab peb sab ua rau lawv lub neej yooj yim dua.
Daim duab yog ua raws li cov hauv qab no. 1 yog sau rau ntawm daim duab peb sab saum toj, thiab nyob rau hauv txhua kab tom ntej nws dhau los ua ib tus lej ntxiv, 1 muab tso rau ntawm cov npoo, thiab nruab nrab ntawm kab yog sau nrog cov lej ntawm ob tus lej nyob ib sab ntawm tus dhau los.
Thaum koj saib cov lus piav qhia, txhua yam yuav pom tseeb.
Tau kawg, kev siv cov lej hauv kev lej tsis txwv rau cov piv txwv muab, cov neeg paub ntau tshaj plaws.
