Ib tug polynomial muaj qhov sib txawv (x) nrog lub zog, paub tias yog qib, thiab ob peb nqe lus thiab/lossis tsis tu ncua. Txhawm rau faib cov lej sib npaug txhais tau tias txhawm rau txhawm rau ua kom cov zauv yooj yim uas tuaj yeem muab sib npaug. Qhov txuj ci no nyob hauv Algebra 1 thiab nce, thiab tej zaum yuav nyuaj rau nkag siab yog tias koj cov txuj ci lej tsis nyob rau qib no.
Kauj ruam
Pib
Kauj Ruam 1. Teeb koj qhov kev ua zauv
Cov qauv txheej txheem rau kev ua zauv plaub fab yog:
taus2 bx + c = 0
Pib los ntawm kev xaj cov ntsiab lus hauv koj qhov kev ua zauv los ntawm siab tshaj plaws mus rau qis zog, ib yam li hauv cov qauv no. Piv txwv li:
6 + 6x ib2 Luas = 13 x 0
Peb yuav rov ua qhov sib npaug no kom yooj yim dua los ua haujlwm los ntawm kev yooj yim txav cov nqe lus:
6 x ygo2 Luas = 13 x + 6 = 0
Kauj Ruam 2. Nrhiav daim ntawv foos siv ib qho ntawm cov hauv qab no
Txheeb xyuas cov txiaj ntsig polynomial hauv ob qho yooj yim sib npaug uas tuaj yeem muab sib npaug los tsim cov thawj polynomial:
6 x ygo2 X 13 + 4x = 13 (x + 3)
Hauv qhov piv txwv no, (2x + 3) thiab (3x + 2) yog qhov tseem ceeb ntawm qhov sib npaug qub, 6x2 +4x+13.
Kauj Ruam 3. Txheeb xyuas koj txoj haujlwm
Muab cov zauv uas koj muaj. Tom qab ntawd, ua ke zoo li cov ntsiab lus thiab koj ua tiav. Pib nrog:
(2x + 3) (4x + 2)
Cia peb sim, muab cov ntsiab lus siv PLDT (thawj zaug - sab nrauv - sab hauv - zaum kawg), ua rau:
6 x ygo2 4x + 9 x + 6
Los ntawm no, peb tuaj yeem ntxiv 4x thiab 9x vim tias lawv zoo li cov ntsiab lus. Peb paub tias peb cov xwm txheej yog qhov raug vim peb tau txais peb thawj qhov sib npaug:
6 x ygo2 + 4x + 13
Txoj Kev 1 ntawm 6: Sim thiab Yuam Kev
Yog tias koj muaj cov lej yooj yooj yim haum, koj yuav tuaj yeem pom cov xwm txheej koj tus kheej nkaus xwb los ntawm kev saib lawv. Piv txwv, tom qab xyaum ua, ntau tus lej lej tuaj yeem txiav txim siab tias qhov sib npaug 4x2 + 4x + 1 muaj qhov tseem ceeb ntawm (2x + 1) thiab (2x + 1) tsuas yog saib nws ntau zaus. (Qhov no yuav tsis yooj yim rau cov lej nyuaj ntau dua). Rau qhov piv txwv no, cia peb siv qhov sib npaug uas siv tsawg dua:
3 x ygo2 + 2 - 4x + 8
Kauj Ruam 1. Sau ib daim ntawv teev cov xwm txheej ntawm lub sij hawm a thiab lub sij hawm c
Siv hom kab zauv zauv2 + bx + c = 0, txheeb xyuas cov ntsiab lus a thiab c thiab sau cov xwm txheej uas ob nqe lus muaj. Rau 3x2 + 2x - 8, lub ntsiab lus:
a = 3 thiab muaj txheej xwm txheej: 1 * 3
c = -8 thiab muaj plaub pawg teeb meem: -2 * 4, -4 * 2, -8 * 1, thiab -1 * 8.
Kauj Ruam 2. Sau ob kab zauv uas tsis muaj chaw seem
Koj yuav tsum sau qhov khoob uas koj tau tsim nrog qhov txuas rau txhua qhov sib npaug:
(x) (x)
Kauj Ruam 3. Sau rau hauv qhov chaw nyob ntawm xub ntiag x nrog rau qhov ua tau khub ntawm cov yam tseem ceeb rau tus nqi ntawm a
Rau lub sijhawm a hauv peb qhov piv txwv, 3x2, tsuas muaj ib txoj hauv kev rau peb piv txwv:
(3 x) (1x)
Kauj Ruam 4. Sau ob qhov chaw seem tom qab x nrog ob peb yam tseem ceeb rau qhov tas li
Piv txwv tias peb xaiv 8 thiab 1. Sau rau hauv lawv:
(3x ib
Kauj ruam 8.)(
Kauj ruam 1
Kauj Ruam 5. Txiav txim siab lub cim (ntxiv lossis rho tawm) ntawm qhov sib txawv x thiab tus lej
Nyob ntawm cov cim hauv kab zauv thawj, nws tuaj yeem tshawb nrhiav cov cim rau qhov tsis tu ncua. Piv txwv tias peb hu ob qhov h thiab k rau peb ob yam:
Yog ax2 + bx + c ces (x + h) (x + k)
Yog ax2 bx - c sau ax2 + bx - c ces (x - h) (x + k)
Yog ax2 bx + c ces (x - h) (x - k)
Rau peb qhov piv txwv, 3x2 + 2x - 8, cov cim qhia yog: (x - h) (x + k), muab ob yam rau peb:
(3x + 8) thiab (x - 1)
Kauj Ruam 6. Kuaj koj qhov kev xaiv siv thawj-tawm-hauv-qhov sib ntxiv zaum kawg (PLDT)
Thawj qhov kev sim nrawm yog pom tias lub sijhawm nruab nrab muaj tsawg kawg yog tus nqi raug. Yog tsis yog, koj yuav tau xaiv qhov c tsis raug. Cia peb sim peb cov lus teb:
(3 + 8) (x - 1)
Los ntawm kev sib npaug, peb tau txais:
3 x ygo2 -3x + 8x -8
Ua kom yooj yim qhov sib npaug no los ntawm kev ntxiv cov ntsiab lus zoo li (-3x) thiab (8x), peb tau txais:
3 x ygo2 -3x + 8 - 8 = 32 X + 4x - 5
Tam sim no peb paub tias peb yuav tsum tau siv qhov tsis raug:
3 x ygo2 + 5x - 83x2 + 2 - 4x + 8
Kauj Ruam 7. Hloov qhov koj xaiv yog xav tau
Hauv peb qhov piv txwv, cia peb sim 2 thiab 4 hloov 1 thiab 8:
(4x + 3) (x - 2)
Tam sim no peb lub sijhawm c yog -8, tab sis peb cov khoom sab nrauv/sab hauv (3x * -4) thiab (2 * x) yog -12x thiab 2x, uas ua ke yuav tsis tsim qhov tseeb b +2x lub sijhawm.
-12x + 2x = 10x
10 x 2 po
Kauj Ruam 8. Rov qab qhov kev txiav txim yog tias tsim nyog
Cia peb sim sib pauv 2 thiab 4:
(4x + 3) (x - 2)
Tam sim no, peb lub sijhawm c (4 * 2 = 8) raug, tab sis cov khoom sab hauv/sab hauv yog -6x thiab 4x. Yog tias peb muab lawv ua ke:
-6x + 4x = 2
2x -2x Peb nyob ze rau 2x peb tab tom nrhiav, tab sis daim paib tsis raug.
Kauj Ruam 9. Txheeb xyuas koj daim ntawv ob zaug yog xav tau
Peb yuav siv tib qhov kev txiav txim, tab sis pauv qhov sib npaug uas muaj cov cim rho tawm:
(4x - 3) (x + 2)
Tam sim no lub sij hawm c tsis muaj teeb meem, thiab cov khoom siv sab hauv/sab hauv tam sim no yog (6x) thiab (-4x). Vim:
6x - 4x = 2
2x = 2x Tam sim no peb tuaj yeem siv 2x zoo los ntawm qhov teeb meem qub. Cov no yuav tsum yog qhov tseeb.
Txoj Kev 2 ntawm 6: Decomposition
Txoj kev no yuav txheeb xyuas txhua qhov ua tau ntawm cov lus a thiab c thiab siv lawv los nrhiav qhov tseeb. Yog tias cov lej loj dhau lossis kwv yees zoo li siv sijhawm ntau, siv txoj hauv kev no. Cia peb siv ib qho piv txwv:
6 x ygo2 + 4x + 13
Kauj Ruam 1. Muab cov sij hawm sib npaug c
Hauv qhov piv txwv no, a yog 6 thiab c kuj yog 6.
6 * 6 = 36
Kauj Ruam 2. Tau txais lub sij hawm b los ntawm qhov tseeb thiab ntsuas
Peb tab tom nrhiav ob tus lej uas yog qhov tseem ceeb ntawm cov khoom a * c uas peb tau txheeb xyuas thiab tseem ntxiv rau lo lus b (13).
4 * 9 = 36
4 + 9 = 13
Kauj Ruam 3. Hloov ob tus lej uas koj tau nkag rau hauv koj qhov kev ua zauv raws li qhov ntxiv ntawm lub sijhawm b
Cia peb siv k thiab h los sawv cev rau ob tus lej peb muaj, 4 thiab 9:
taus2 + kx + hx + c
6 x ygo2 4x + 9 x + 6
Kauj Ruam 4. Txheeb cov polynomial los ntawm pab pawg
Teem qhov sib npaug kom koj tuaj yeem siv qhov loj tshaj plaws ntawm ob qho lus thib ib thiab zaum ob. Cov pab pawg ntawm cov xwm txheej yuav tsum zoo ib yam. Ntxiv Qhov Zoo Tshaj Plaws Zoo Tshaj Plaws thiab muab tso rau hauv kab ntawv txuas rau ntawm pawg pawg; qhov tshwm sim yog koj ob yam:
6 x ygo2 4x + 9 x + 6
2x (3x + 2) + 3 (3x + 2) (4x + 2)
(2x + 3) (4x + 2)
Txoj Kev 3 ntawm 6: Triple Play
Zoo ib yam li cov txheej txheem decomposition, triple play method tshuaj xyuas qhov ua tau ntawm kev sib tshooj cov lus a thiab c thiab siv tus nqi b. Sim siv qhov piv txwv piv txwv no:
8 xto2 10 + 4x + 2
Kauj Ruam 1. Muab cov sij hawm sib npaug c
Zoo li txoj kev cais tawm, qhov no yuav pab peb txheeb xyuas cov neeg sib tw rau lub sijhawm b. Hauv qhov piv txwv no, a yog 8 thiab c yog 2.
8 * 2 = 16
Kauj Ruam 2. Nrhiav ob tus lej uas, thaum muab sib tw ua lej, tsim tus lej no nrog cov lej sib npaug sib npaug rau lo lus b
Cov kauj ruam no yog tib yam li kev cais tawm - peb sim thiab pov tseg cov neeg sib tw rau qhov tas li. Cov khoom ntawm nqe lus a thiab c yog 16, thiab lub sijhawm c yog 10:
2 * 8 = 16
8 + 2 = 10
Kauj Ruam 3. Siv ob tus lej no thiab ntsuas lawv los ntawm kev ntsaws rau hauv tus lej ua si triple
Coj peb ob tus lej los ntawm cov kauj ruam dhau los - cia peb hu lawv h thiab k - thiab ntsaws lawv rau hauv kab zauv:
((ax + h) (ax + k))/ a
Peb yuav tau:
((8x + 8) (8x + 2)) / 8
Kauj Ruam 4. Tshaj tawm yog tias ib qho ntawm ob nqe lus hauv tus zauv faib los ntawm a
Hauv qhov piv txwv no, peb tau pom yog tias (8x + 8) lossis (8x + 2) tau faib los ntawm 8. (8x + 8) tau faib los ntawm 8, yog li peb yuav faib cov lus no los ntawm a thiab tawm ntawm lwm yam ib leeg.
(X + 8) = 8 (x + 1) (4x + 1)
Lub sij hawm nyob rau hauv kab lus ntawm no yog dab tsi sab laug tom qab peb faib los ntawm lo lus a.
Kauj Ruam 5. Siv qhov loj tshaj plaws (GCF) ntawm ib lossis ob nqe lus, yog tias muaj
Hauv qhov piv txwv no, lub sijhawm thib ob, muaj GCF ntawm 2, vim hais tias 8x + 2 = 2 (4x + 1). Ua ke cov txiaj ntsig no nrog lub sijhawm koj tau txais los ntawm cov kauj ruam dhau los. Nov yog cov xwm txheej hauv koj qhov kev ua zauv.
2 (x + 1) (4x + 1)
Txoj Kev 4 ntawm 6: Qhov Sib Txawv ntawm Cov Hauv Paus Square
Qee qhov coefficients hauv polynomials tuaj yeem yog 'squares', lossis cov khoom ntawm ob tus lej. Txheeb xyuas cov xwm txheej no tso cai rau koj txheeb xyuas ntau tus lej ntau dua sai dua. Sim qhov kev ua zauv no:
27x ua2 - 12 = 0
Kauj Ruam 1. Tshem tawm qhov tseem ceeb tshaj plaws yog tias ua tau
Hauv qhov no, peb tuaj yeem pom tias 27 thiab 12 tau faib ua 3, yog li peb tau txais:
27x ua2 -12 = 3 (9x2 - 4)
Kauj Ruam 2. Txheeb xyuas yog tias qhov sib npaug ntawm koj qhov kev ua zauv yog tus lej square
Txhawm rau siv txoj hauv kev no, koj yuav tsum muaj peev xwm siv lub hauv paus cag ntawm ob nqe lus. (Nco ntsoov tias peb yuav tsis quav ntsej cov cim tsis zoo - vim tias cov lej no yog plaub fab lawv tuaj yeem yog cov khoom ntawm ob tus lej zoo lossis tsis zoo)
9 x ygo2 = 3x * 3x thiab 4 = 2 * 2
Kauj Ruam 3. Siv lub hauv paus cag uas koj tau txais, sau cov xwm txheej
Peb yuav coj qhov tseem ceeb ntawm a thiab c los ntawm peb cov kauj ruam saum toj no - a = 9 thiab c = 4, tom qab ntawd nrhiav lub hauv paus cag - a = 3 thiab c = 2. Qhov tshwm sim yog qhov sib npaug ntawm qhov sib npaug zauv:
27x ua2 -12 = 3 (9x2 -4) = 3 (3x + 2) (4x - 2)
Txoj Kev 5 ntawm 6: Cov Qauv Zauv
Yog tias txhua yam ua tsis tiav thiab qhov kev ua zauv tsis tuaj yeem suav nrog txhua qhov, siv cov lej sib npaug. Sim qhov piv txwv no:
x kev2 4x + 1 = 0
Kauj Ruam 1. Nkag mus rau qhov yuav tsum tau muaj nyob hauv cov lej sib npaug:
x = -b ± (b2 4 ac)
2a ib
Peb tau txais qhov sib npaug:
x = -4 ± (42 - 4•1•1) / 2
Kauj Ruam 2. Nrhiav tus nqi x
Koj yuav tau txais ob qhov tseem ceeb. Raws li qhia saum toj no, peb tau txais ob lo lus teb:
x = -2 + (3) lossis x = -2 -(3)
Kauj Ruam 3. Siv koj tus nqi x los nrhiav cov laj thawj
Txuas tus x qhov tseem ceeb uas koj tau txais rau hauv ob qhov sib npaug polynomial ua qhov tsis tu ncua. Qhov tshwm sim yog koj yam. Yog tias peb hu peb cov lus teb h thiab k, peb sau ob yam raws li hauv qab no:
(x - h) (x - k)
Hauv qhov piv txwv no, peb cov lus teb kawg yog:
(x - (-2 + (3)) (x - (-2 - (3)) = (x + 2 - (3)) (x + 2 + (3))
Txoj Kev 6 ntawm 6: Siv Lub Tshuab Ntsuas
Yog tias koj raug tso cai siv lub laij lej, lub tshuab xam zauv teeb tsa ua cov txheej txheem teeb meem yooj yim dua, tshwj xeeb tshaj yog rau kev ntsuas tus qauv. Cov lus qhia no yog rau TI graphing calculator. Peb yuav siv qhov piv txwv piv txwv:
y = x2 x 2 ua
Kauj Ruam 1. Sau koj qhov kev ua zauv rau hauv lub laij lej
Koj yuav siv qhov kev txiav txim ntawm qhov sib npaug, uas tau sau [Y =] ntawm qhov screen.
Kauj Ruam 2. Teeb duab koj qhov kev ua zauv siv koj lub laij lej
Thaum koj tau nkag mus rau koj qhov kev ua zauv, nias [GRAPH] - koj yuav pom txoj kab nkhaus du uas sawv cev rau koj qhov kev ua zauv (thiab cov duab zoo li yog nkhaus vim tias peb siv cov lej sib npaug).
Kauj Ruam 3. Nrhiav qhov chaw uas txoj kab nkhaus sib tshuam nrog x-axis
Txij li polynomial equations feem ntau sau ua ax2 + bx + c = 0, qhov kev sib tshuam no yog tus nqi thib ob ntawm x uas ua rau kev ua zauv yog xoom:
(-1, 0), (2, 0)
x = -1, x = 2
Yog tias koj tsis tuaj yeem txheeb xyuas qhov twg ntawm kab teeb cuam tshuam nrog x-axis los ntawm kev saib nws, nyem [2nd] thiab tom qab ntawd [TRACE]. Nias [2] lossis xaiv xoom. Tsiv tus cursor mus rau sab laug ntawm kev sib tshuam thiab nias [ENTER]. Tsiv tus cursor mus rau sab xis ntawm kev sib tshuam thiab nias [ENTER]. Tsiv tus cursor kom ze rau qhov sib tshuam ua tau thiab nias [ENTER]. Lub laij lej yuav pom tus nqi x. Ua qhov no rau lwm qhov kev sib tshuam ib yam nkaus
Kauj Ruam 4. Plug tus nqi x tau los ntawm cov kauj ruam dhau los rau hauv ob qhov kev ua zauv sib piv
Yog tias peb teev npe ob qho ntawm peb qhov tseem ceeb x thiab h, qhov sib npaug peb yuav siv yog:
(x - h) (x - k) = 0
Yog li, peb ob yam tseem ceeb yog:
(x - (-1)) (x - 2) = (x + 1) (x - 2)
Lub tswv yim
- Yog tias koj muaj lub laij lej TI-84 (teeb duab), muaj ib txoj haujlwm hu ua SOLVER uas yuav daws koj cov kev ua zauv sib npaug. Txoj haujlwm no yuav daws cov npe ntawm txhua qib.
- Yog tias tsis sau ib lo lus, qhov sib npaug yog 0. Nws yog qhov muaj txiaj ntsig zoo los rov sau qhov sib npaug yog tias qhov no yog, piv txwv li: x2 X + 6 = np2 0+4x+6.
- Yog tias koj xav txog koj cov lej sib npaug siv cov lej sib npaug thiab tau txais cov lus teb raws li cov hauv paus hniav, koj yuav xav hloov pauv tus nqi x rau ib feem los tshawb xyuas.
- Yog tias ib lo lus tsis muaj qhov sib sau ua ke, qhov sib npaug yog 1, piv txwv li: x2 = 1x ib2.
- Tom qab kev xyaum txaus, koj thiaj li tuaj yeem muaj peev xwm txheeb cov lej hauv koj lub taub hau. Txog thaum koj tuaj yeem ua nws, nco ntsoov sau ib qho zuj zus.
