Trinomial yog kev qhia ua lej suav nrog peb nqe lus. Feem ntau yuav, koj yuav pib kawm paub yuav ua li cas ua ib feem plaub npaug trinomial, lub ntsiab lus trinomial sau hauv daim ntawv ax2 bx + c. Muaj qee qhov txuj ci los kawm, uas tuaj yeem siv rau ntau hom sib txawv ntawm cov lej trinomials, tab sis koj tuaj yeem siv lawv tau zoo dua thiab nrawm dua nrog kev xyaum. Kev txiav txim siab ntau dua polynomials, nrog cov ntsiab lus zoo li x3 sau x4, tsis tuaj yeem daws tau ib yam nkaus, tab sis koj tuaj yeem siv tau qhov yooj yim qhov tseeb lossis hloov pauv los ua qhov teeb meem uas tuaj yeem daws tau zoo ib yam li lwm tus lej sib npaug.
Kauj ruam
Txoj Kev 1 ntawm 3: Factoring x2 bx + c
Kauj Ruam 1. Kawm PLDT cov lej
Tej zaum koj yuav tau kawm paub yuav ua li cas los sib tw PLDT, lossis "Ua Ntej, Sab Nraud, Hauv, Xeem" kom muab cov lus sib piv zoo li (x+2) (x+4). Nws muaj txiaj ntsig kom paub yuav ua li cas sib npaug ua haujlwm ua ntej peb ntsuas:
- Multiply pab pawg neeg Thawj: (x kev+2)(x kev+4) = x kev2 + _
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Multiply pab pawg neeg Sab nraum: (x kev(2) (x+x+4x
Kauj ruam 4.) = xwm2+ 4x ua + _
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Multiply pab pawg neeg Hauv: (x+ os
Kauj ruam 2.)(x kev4) x = np24x+x 2x ua + _
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Multiply pab pawg neeg Kawg: (x+ os
Kauj ruam 2.) (xwm
Kauj ruam 4.) = xwm24x+2 x
Kauj ruam 8.
- Ua kom yooj yim: x24x+x+8 = x2X+4x+8
Kauj Ruam 2. Nkag siab txog qhov tseeb
Thaum koj muab ob npaug zauv siv PLDT txoj hauv kev, koj tau txais trinomial (kev qhia nrog peb nqe lus) hauv daim ntawv x2+ b x+ c, qhov twg a, b, thiab c yog cov lej zoo tib yam. Yog tias koj pib nrog qhov sib npaug uas muaj tib daim ntawv, koj tuaj yeem muab nws rov qab rau hauv ob lub zauv.
- Yog tias cov kab zauv tsis tau sau rau hauv qhov kev txiav txim no, rov kho qhov sib npaug kom lawv muaj daim ntawv xaj no. Piv txwv li, sau dua tshiab 3 - x 10 + 4x2 Ua x kev2 X + 10 - 4x.
- Vim tias lub zog siab tshaj yog 2 (x2, hom kev qhia no hu ua quadratic.
Kauj Ruam 3. Tawm qhov chaw seem rau cov lus teb hauv daim ntawv PLDT sib npaug
Txog tam sim no, tsuas yog sau (_ _)(_ _) qhov twg koj yuav sau cov lus teb. Peb yuav sau nws thaum ua haujlwm rau nws
Tsis txhob sau + lossis - nruab nrab ntawm cov nqe lus khoob vim tias peb tseem tsis tau paub lub cim raug
Kauj Ruam 4. Sau thawj cov lus
Rau cov teeb meem yooj yim, thawj lub sijhawm ntawm koj trinomial tsuas yog x2, cov ntsiab lus hauv Thawj txoj haujlwm ib txwm muaj x kev thiab x kev. Nov yog cov xwm txheej ntawm lo lus x2 vim x x x = x2.
- Peb qhov piv txwv x2 + 3x - 10 pib nrog x2, yog li peb tuaj yeem sau:
- (x _) (x _)
- Peb yuav ua haujlwm ntawm cov teeb meem nyuaj hauv ntu tom ntej, suav nrog trinomials pib nrog cov lus zoo li 6x2 lub -x2. Lub sijhawm no, ua raws cov lus nug piv txwv no.
Kauj Ruam 5. Siv qhov kev txiav txim kom kwv yees cov lus kawg
Yog tias koj rov qab thiab nyeem cov kauj ruam yuav ua li cas los sib tw PLDT, koj yuav pom tias kev sib tshooj Cov Lus Kawg yuav tsim lub sijhawm kawg hauv cov lej (cov ntsiab lus uas tsis muaj x). Yog li txhawm rau txhawm rau, peb yuav tsum nrhiav ob tus lej uas thaum sib tw yuav tsim lub sijhawm kawg.
- Hauv peb qhov piv txwv x2 + 3x - 10, lub sijhawm kawg yog -10.
- Yam dab tsi ntawm -10? Tus lej twg yog sib npaug -10?
- Muaj ntau qhov ua tau: -1 zaug 10, 1 zaug -10, -2 zaug 5, lossis 2 zaug -5. Sau cov khub no rau qee qhov kom nco tau lawv.
- Tsis txhob hloov peb cov lus teb tam sim no. Peb cov lus teb yuav tsum zoo li no: (x _) (x _).
Kauj Ruam 6. Ntsuas qhov muaj peev xwm uas haum rau Cov Khoom Sab Hauv thiab Sab Hauv
Peb tau nqes cov lus kawg mus rau qee qhov muaj peev xwm. Siv cov txheej txheem sim los ntsuas txhua qhov ua tau, muab cov lus sab hauv thiab sab hauv thiab sib piv cov khoom lag luam nrog peb trinomial. Piv txwv li:
- Peb qhov teeb meem qub muaj lo lus "x" ntawm 3x, yog li peb qhov kev xeem yuav tsum ua raws lub sijhawm no.
- Kuaj -1 thiab 10: (x -1) (x+10). Sab nrauv + Sab hauv = 10x - x = 9x. Tsis ncaj ncees lawm.
- Kuaj 1 thiab -10: (x+1) (x -10). -10x + x = -9x. Qhov no tsis yog lawm. Qhov tseeb, yog tias koj sim -1 thiab 10, koj yuav pom tias 1 thiab -10 yog qhov sib txawv ntawm cov lus teb saum toj no: -9x tsis yog 9x.
- Kuaj -2 thiab 5: (x -2) (x+5). 5 - 4x = 3 x Qhov tshwm sim sib xws rau thawj polynomial, yog li ntawm no yog cov lus teb raug: (x-2) (x+5).
- Hauv cov xwm txheej yooj yim zoo li no, yog tias koj tsis muaj qhov tas li nyob rau hauv ntej ntawm lub sijhawm x2, koj tuaj yeem siv txoj hauv kev nrawm: tsuas yog ntxiv ob qhov tseem ceeb thiab tso "x" tom qab nws (-2+5 → 3x). Txawm li cas los xij, txoj hauv kev no tsis ua haujlwm rau teeb meem nyuaj dua, yog li nws zoo dua kom nco qab "txoj hauv kev ntev" piav qhia saum toj no.
Txoj Kev 2 ntawm 3: Factoring Ntau Yam Trinomials
Kauj Ruam 1. Siv qhov kev txiav txim yooj yim los ua teeb meem nyuaj dua
Piv txwv li, koj yuav tsum xav txog 3 x ygo2 + 9 x 30 po. Nrhiav tus lej uas tuaj yeem suav tag nrho peb nqe lus ("qhov zoo tshaj plaws" lossis GCF). Hauv qhov no, GCF yog 3:
- 3 x ygo2 = (3) (x2)
- 9 = 4x (x)
- -30 = (3)(-10)
- Yog, 3x2 + 9x - 30 = (3) (x2+4x -10). Peb tuaj yeem txiav tawm qhov trinomial tshiab siv cov kauj ruam hauv ntu saum toj no. Peb qhov lus teb kawg yuav yog (3) (x-2) (x+5).
Kauj Ruam 2. Saib cov teeb meem nyuaj dua
Qee zaum, qhov teeb meem tuaj yeem cuam tshuam nrog qhov sib txawv, lossis koj yuav xav tau ob peb zaug los nrhiav qhov yooj yim tshaj plaws tau hais tawm. Nov yog qee qhov piv txwv:
- 2x ua2y + 14xy + 24y = (2 xyoo)(xwm2 (X 7x + 4x + 12)
- x kev4 X + 11 po3 - 26x ib2 = (xwm2)(xwm2 +11x - 26)
- -x ib2 6x + 9 = np (-1)(xwm2 -6x + 9)
- Tsis txhob hnov qab rov ua dua lub trinomial tshiab, siv cov kauj ruam hauv Txoj Kev 1. Txheeb xyuas koj txoj haujlwm thiab saib cov piv txwv ntawm cov teeb meem zoo sib xws hauv cov lus nug piv txwv nyob hauv qab ntawm nplooj ntawv no.
Kauj Ruam 3. Daws teeb meem nrog tus lej ua ntej ntawm x2.
Qee qhov trinomials plaub npaug tsis tuaj yeem raug txo kom yooj yim yam teeb meem. Kawm paub daws teeb meem zoo li 3x2 + 10x + 8, tom qab ntawd xyaum ntawm koj tus kheej nrog cov lus nug piv txwv hauv qab ntawm nplooj ntawv no:
- Teem peb cov lus teb kom yog: (_ _)(_ _)
- Peb cov lus "Thawj" yuav txhua tus muaj ib qho x, thiab muab lawv sib npaug 3x2. Tsuas muaj ib qho ua tau: (3 x _) (x _).
- Teev cov xwm txheej ntawm 8. Qhov sib txawv yog 1 npaug 8 lossis 2 npaug 4.
- Kuaj qhov ua tau no siv Cov Txheej Txheem Sab Hauv thiab Sab Hauv. Nco ntsoov tias qhov kev txiav txim ntawm qhov tseem ceeb heev vim tias Lub Sijhawm Sab Nrauv tau muab sib npaug 3 x tsis yog x. Sim txhua qhov ua tau kom txog thaum koj Tawm Tawm+Hauv = 10x (los ntawm qhov teeb meem qub):
- (3x+1) (x+8) → 24x+x = 25x tsis muaj
- (3x+8) (x+1) → 3x+8x = 11x tsis muaj
- (3x+2) (x+4) → 12x+2x = 14x tsis muaj
- (3x+4) (x+2) → 6x+4x = 10x yog. Qhov no yog qhov tseeb.
Kauj Ruam 4. Siv kev hloov pauv rau kev txiav txim siab trinomials
Koj phau ntawv ua lej yuav ua rau koj xav tsis thoob nrog qhov sib npaug nrog lub zog loj, xws li x4, txawm tias tom qab koj siv cov lej yooj yim los ua qhov teeb meem yooj yim dua. Sim hloov qhov hloov pauv tshiab uas hloov nws mus rau hauv qhov teeb meem koj paub daws teeb meem. Piv txwv li:
- x kev5X +13 po3X +36 po
- = (x) (x4X +13 po2+36)
- Cia peb tsim qhov hloov pauv tshiab. Cia peb hais tias y = x2 thiab muab tso rau hauv:
- (x) (y2+13y+36)
- = (x) (y+9) (y+4). Tam sim no, hloov nws rov qab mus rau qhov pib sib txawv:
- = (x) (x2+9) (xwv2+4)
- = (x) (x ± 3) (x ± 2)
Txoj Kev 3 ntawm 3: Txheeb Xyuas Cov Ntaub Ntawv Tshwj Xeeb
Kauj Ruam 1. Nrhiav tus lej tseem ceeb
Saib seb puas yog qhov tas li hauv thawj lossis thib peb ntawm trinomial yog tus lej tseem ceeb. Tus lej tseem ceeb tsuas yog faib tau ntawm nws tus kheej thiab 1, yog li tsuas muaj ib qho ua tau ntawm ob yam sib txawv.
- Piv txwv li, hauv x2 + 6x + 5, 5 yog tus lej tseem ceeb, yog li binomial yuav tsum yog daim ntawv (_ 5) (_ 1).
- Qhov teeb meem ntawm 3x2+10x+8, 3 yog tus lej tseem ceeb, yog li binomial yuav tsum yog daim ntawv (3x _) (x _).
- Rau cov lus nug 3x2+4x+1, ob qho tib si 3 thiab 1 yog tus lej tseem ceeb, yog li qhov kev daws tau nkaus xwb yog (3x+1) (x+1). (Koj tseem yuav tsum muab tus lej no los tshuaj xyuas koj cov lus teb vim tias qee qhov kev hais tawm tsis tuaj yeem ua qhov tseeb - piv txwv li, 3x2+100x+1 tsis muaj qhov zoo.)
Kauj Ruam 2. Tshawb xyuas seb lub trinomial yog lub xwmfab zoo tshaj plaws
Ib lub xwmfab trinomial zoo meej tuaj yeem muab faib ua ob qhov zoo sib xws, thiab qhov feem ntau yog sau ua (x+1)2 thiab tsis yog (x+1) (x+1). Nov yog qee qhov piv txwv uas zoo li tshwm sim hauv cov lus nug:
- x kev2X 2x+4x+1 = (x+1)2, thiab x2-2x+1 = (x-1)2
- x kev24x+4 = (x+2)2, thiab x2-4x+4 = (x-2)2
- x kev2X+6x+9 = (x+3)2, thiab x2-6x+9 = (x-3)2
- Zoo meej square trinomial hauv daim ntawv x2 + bx + c ib txwm muaj cov ntsiab lus a thiab c uas zoo sib xws (xws li 1, 4, 9, 16, lossis 25) thiab ib lo lus b (zoo lossis tsis zoo) uas yog sib npaug rau 2 (*a * √c).
Kauj Ruam 3. Tshawb xyuas yog tias teeb meem tsis muaj kev daws teeb meem
Tsis yog txhua tus trinomials tuaj yeem raug txiav txim siab. Yog tias koj tsis tuaj yeem ntsuas qhov sib npaug trinomial (ax2+bx+c), siv cov lej sib npaug los nrhiav cov lus teb. Yog tias cov lus teb nkaus xwb yog lub hauv paus cag ntawm tus lej tsis zoo, tsis muaj tus lej tiag daws, tom qab ntawd qhov teeb meem tsis muaj yam tseem ceeb.
Rau cov tsis-square trinomials, siv Eisenstein Criterion, uas tau piav qhia hauv ntu Cov Lus Qhia
Cov Lus Teb thiab Cov Lus Nug Piv Txwv
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Cov lus teb rau "cov teeb meem nyuaj" cov lus nug.
Cov no yog cov lus nug los ntawm "yam nyuaj dua" cov kauj ruam. Peb tau yooj yim daws teeb meem rau qhov yooj yim dua, yog li sim daws lawv siv cov kauj ruam hauv txoj kev 1, tom qab ntawd txheeb xyuas koj txoj haujlwm ntawm no:
- (2 y) (x2 X 7x + 4x + 12) = (x+3) (x+4x)
- (xwm2) (xwm2 (11x - 26) = np (x+13) (x-2)
- (-1) (x2 6x + 9) = (x-3) (x-3) = (x-3)2
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Sim ua qhov teeb meem nyuaj dua qhov teeb meem.
Cov teeb meem no muaj tib yam tseem ceeb hauv txhua lub sijhawm uas yuav tsum tau muab cais ua ntej. Thaiv qhov khoob tom qab kos npe sib npaug kom pom cov lus teb yog li koj tuaj yeem tshawb xyuas koj txoj haujlwm:
- 3 x ygo3+3 x po2-6x = (3x) (x+2) (x-1) thaiv qhov khoob kom pom cov lus teb
- -5x ib3y2X +30 po2y2-25 xyoo2x = (-5xy^2) (x-5) (x-1)
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Xyaum siv cov lus nug. Cov teeb meem no tsis tuaj yeem ua rau sib npaug yooj yim dua, yog li koj yuav tsum nrhiav cov lus teb hauv daim ntawv (_x + _) (_ x + _) siv kev sim thiab ua yuam kev:
- 2x ua2+3x-5 = (2x+5) (x-1) thaiv kom pom cov lus teb
- 9 x ygo26x+1 = (3x+1) (3x+1) = (3x+1)2 (Hint: Koj yuav xav sim ntau dua ib qho khub rau 9x.)
Lub tswv yim
- Yog tias koj tsis tuaj yeem txiav txim siab yuav ua li cas txhawm rau faib plaub npaug trinomial (ax2+bx+c), koj tuaj yeem siv cov lej sib npaug los nrhiav x.
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Thaum koj tsis tas yuav paub yuav ua li cas, koj tuaj yeem siv Eisenstein Cov Qauv los txiav txim siab sai yog tias muaj ntau tus lej tsis tuaj yeem ua yooj yim dua thiab ua piv txwv. Tus txheej txheem no siv rau txhua tus lej tab sis yog qhov zoo tshaj plaws siv rau trinomials. Yog tias muaj tus lej tseem ceeb p uas faib ob nqe lus dhau los sib npaug thiab txaus siab rau cov xwm txheej hauv qab no, tom qab ntawd cov lej tsis tuaj yeem yooj yim:
- Cov ntsiab lus tsis tu ncua (tsis muaj qhov txawv txav) yog p ntau tiam sis tsis yog sib npaug ntawm p2.
- Lub prefix (piv txwv li, nyob rau hauv ax2+bx+c) tsis yog ntau ntawm p.
- Ej.: 14 x2 +45x +51 tsis tuaj yeem yooj yim vim tias muaj tus lej tseem ceeb (3) uas tuaj yeem faib los ntawm ob qho tib si 45 thiab 51, tab sis tsis faib los ntawm 14, thiab 51 tsis tuaj yeem faib ua 32.
Ceeb toom
Thaum qhov no muaj tseeb rau kev sib tw trinomials plaub, trinomial uas tuaj yeem ua qhov tseeb tsis tas yuav yog cov khoom ntawm ob lub zauv. Piv txwv li, x4 + 105x + 46 = (x2 (X 5x + 4x + 5)2 -5x + 23).
