Hauv kev ua lej, qhov tseeb yog ib txoj hauv kev nrhiav cov lej lossis kab lus uas thaum sib tw ua ke yuav tsim tus lej lossis kab zauv. Factoring yog cov txuj ci muaj txiaj ntsig los kawm los daws teeb meem lej lej yooj yim; lub peev xwm los ntsuas qhov zoo, dhau los ua qhov tseem ceeb thaum cuam tshuam nrog cov lej sib npaug thiab lwm yam ntaub ntawv ntawm polynomials. Factoring tuaj yeem siv los ua kom yooj yim piav qhia algebraic los ua lawv cov kev daws teeb meem yooj yim dua. Factoring tseem tuaj yeem muab lub peev xwm rau koj tshem tawm qee yam lus teb tau, sai dua li daws lawv tus kheej.
Kauj ruam
Txoj Kev 1 ntawm 3: Tus lej zauv thiab kab zauv yooj yim Algebraic
Kauj Ruam 1. Nkag siab lub ntsiab lus ntawm qhov tseeb thaum siv rau tus lej
Factoring yog lub tswv yim yooj yim, tab sis hauv kev coj ua, nws tuaj yeem nyuaj thaum siv rau cov kab zauv nyuaj. Yog li ntawd, nws yog qhov yooj yim tshaj plaws los mus rau lub tswv yim ntawm kev txiav txim siab los ntawm kev pib nrog cov lej yooj yim, tom qab ntawd mus rau qhov sib npaug yooj yim, ua ntej thaum kawg txav mus rau cov ntawv thov nyuaj dua. Cov lej ntawm tus lej yog tus lej uas thaum sib tw ua tus lej. Piv txwv, cov lej ntawm 12 yog 1, 12, 2, 6, 3, thiab 4, vim tias 1 × 12, 2 × 6, thiab 3 × 4 yog sib npaug rau 12.
- Lwm txoj hauv kev xav txog nws yog tias cov lej ntawm tus lej yog tus lej uas tuaj yeem faib sib npaug rau tus lej.
-
Koj tuaj yeem pom txhua qhov xwm txheej ntawm tus lej 60? Peb siv tus lej 60 rau ntau lub hom phiaj (feeb hauv ib teev, vib nas this hauv ib feeb, thiab lwm yam) vim tias nws tuaj yeem faib los ntawm ntau lwm tus lej.
Qhov tseem ceeb ntawm 60 yog 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, thiab 60
Kauj Ruam 2. Nkag siab tias cov kab lus sib txawv kuj tuaj yeem ua qhov tseeb
Ib yam li tus lej lawv tus kheej tuaj yeem raug txiav txim siab, qhov sib txawv nrog tus lej coefficients tseem tuaj yeem ua qhov sib txawv. Txhawm rau ua qhov no, tsuas yog nrhiav cov xwm txheej ntawm qhov sib txawv coefficients. Paub txog kev ntsuas tus lej sib txawv yog qhov muaj txiaj ntsig zoo rau kev yooj yim sib npaug algebraic cuam tshuam nrog qhov sib txawv ntawd.
-
Piv txwv li, qhov sib txawv 12x tuaj yeem sau ua cov khoom lag luam ntawm cov xwm txheej 12 thiab x. Peb tuaj yeem sau 12x ua 3 (4x), 2 (6x), thiab lwm yam, siv yam twg ntawm 12 ua haujlwm zoo tshaj plaws rau peb lub hom phiaj.
Peb tseem tuaj yeem cuam tshuam 12x ntau zaus. Hauv lwm lo lus, peb tsis tas yuav tsum nres ntawm 3 (4x) lossis 2 (6x) - peb tuaj yeem cuam tshuam 4x thiab 6x los tsim 3 (2 (2x) thiab 2 (3 (2x)) Tau kawg, ob kab lus no yog sib npaug
Kauj Ruam 3. Thov faib cov khoom ntiag tug ntawm kev sib npaug rau zauv zauv zauv zauv lej
Siv koj qhov kev paub ntawm yuav ua li cas faib ob tus lej thiab tus lej sib txawv nrog cov coefficients, koj tuaj yeem ua kom yooj yim sib npaug algebraic yooj yim los ntawm kev nrhiav cov xwm txheej uas cov lej thiab cov lej sib koom hauv algebraic equations. Feem ntau, txhawm rau ua kom yooj yim dua qhov sib npaug, peb sim nrhiav qhov loj tshaj plaws. Txoj kev yooj yim no tuaj yeem ua tau vim yog faib cov cuab yeej ntawm kev sib npaug, uas siv rau txhua tus lej a, b, thiab c. a (b + c) = ab + ac.
- Wb sim ua piv txwv lo lus nug. Txhawm rau txheeb xyuas qhov sib npaug algebraic 12x + 6, ua ntej, cia peb sim nrhiav qhov zoo tshaj plaws ntawm 12x thiab 6. 6 yog tus lej loj tshaj uas tuaj yeem faib sib npaug 12x thiab 6, yog li peb tuaj yeem yooj yim sib npaug rau 6 (2x + 1).
- Cov txheej txheem no tseem siv rau kev ua zauv nrog cov lej tsis zoo thiab feem feem. Piv txwv, x/2 + 4, tuaj yeem ua kom yooj yim rau 1/2 (x + 8), thiab -7x + -21 tuaj yeem muab sib piv rau -7 (x + 3).
Txoj Kev 2 ntawm 3: Txheeb cais Quadratic Equations
Kauj Ruam 1. Ua kom ntseeg tau tias qhov sib npaug nyob hauv daim duab plaub (ax2 bx + c = 0).
Quadratic equations muaj daim ntawv ax2 + bx + c = 0, qhov twg a, b, thiab c yog tus lej tsis sib xws thiab tsis sib npaug rau 0 (nco ntsoov tias muaj peev xwm sib npaug 1 lossis -1). Yog tias koj muaj qhov sib npaug uas muaj ib qho sib txawv (x) uas muaj ib lo lus x rau lub zog ntawm ob lossis ntau dua, koj feem ntau txav cov lus no hauv qhov kev ua zauv siv kev ua lej yooj yim kom tau 0 ntawm ob sab ntawm qhov sib npaug kos npe thiab ax2, lwm yam nyob rau lwm sab.
- Piv txwv li, cia peb xav txog qhov sib npaug algebraic. 5 x ygo2 7 - 4x = 92 + x - 18 tuaj yeem ua kom yooj yim rau x2 + 6x + 9 = 0, uas yog daim ntawv square.
- Sib npaug nrog lub zog loj ntawm x, xws li x3, x lus4, lwm yam tsis yog cov zauv sib npaug Cov kab zauv no yog cov kab zauv sib luag, rau plaub lub zog, thiab ntxiv rau, tshwj tsis yog qhov sib npaug tuaj yeem ua kom yooj yim tshem tawm cov lus x nrog lub zog ntau dua 2.
Kauj Ruam 2. Hauv qhov sib npaug ntawm plaub fab, qhov twg a = 1, ua rau (x+d) (x+e), qhov twg d × e = c thiab d+e = b
Yog tias koj qhov sib npaug ntawm plaub fab nyob hauv daim ntawv x2 + bx + c = 0 (hauv lwm lo lus, yog tias qhov sib npaug ntawm lo lus x2 = 1), nws muaj peev xwm (tab sis tsis tau lees paub) tias txoj hauv kev yooj yim yooj yim luv luv tuaj yeem siv los ntsuas qhov sib npaug. Nrhiav ob tus lej uas thaum sib tw muab c thiab ntxiv los tsim b. Tom qab koj tau tshawb rau ob tus lej d thiab e, muab tso rau hauv cov lus hauv qab no: (x+d) (x+e). Ob nqe lus no, thaum sib npaug, muab koj cov lej sib npaug rau koj - hauv lwm lo lus, lawv yog qhov tseem ceeb ntawm koj qhov kev ua zauv sib npaug.
- Piv txwv li, cia peb xav txog qhov ua zauv sib npaug x2 + 5x + 6 = 0. 3 thiab 2 tau muab sib ntxiv muab 6 thiab tseem ntxiv ntxiv muab 5, yog li peb tuaj yeem yooj yim qhov sib npaug no rau (x + 3) (x + 2).
-
Qhov sib txawv me ntsis hauv txoj hauv kev luv luv no nyob hauv qhov sib txawv ntawm qhov zoo sib xws lawv tus kheej:
- Yog hais tias qhov sib npaug ntawm plaub fab nyob hauv daim ntawv x2-bx+c, koj cov lus teb nyob hauv daim ntawv no: (x - _) (x - _).
- Yog tias qhov sib npaug yog nyob hauv daim ntawv x2+ bx + c, koj cov lus teb zoo li no: (x + _) (x + _).
- Yog hais tias qhov sib npaug nyob hauv daim ntawv x2-bx -c, koj cov lus teb nyob hauv daim ntawv (x + _) (x -_).
- Nco tseg: cov lej hauv qhov chaw tuaj yeem yog ib feem lossis cov zauv. Piv txwv li, kab zauv x2 + (21/2) x + 5 = 0 yog muab rau hauv (x + 10) (x + 1/2).
Kauj Ruam 3. Yog tias ua tau, ntsuas los ntawm kev txheeb xyuas
Ntseeg nws lossis tsis ntseeg, rau qhov tsis yooj yim sib npaug sib npaug, ib qho ntawm cov txheej txheem tso cai yog los tshuaj xyuas qhov teeb meem, tom qab ntawd txiav txim siab cov lus teb tau txog thaum koj pom cov lus teb raug. Txoj kev no tseem paub tias yog kev txheeb xyuas los ntawm kev xeem. Yog hais tias qhov sib npaug yog nyob rau hauv daim ntawv ax2+bx +c thiab a> 1, koj cov lus teb yog nyob hauv daim ntawv (dx +/- _) (ex +/- _), qhov twg d thiab e yog qhov tsis tu ncua ntawm cov lej uas thaum sib npaug muab a. Tsis yog d lossis e (lossis ob qho tib si) tuaj yeem yog 1, txawm hais tias nws tsis tas yuav yog. Yog tias ob qho tib si yog 1, koj ib txwm siv txoj hauv kev luv luv piav qhia saum toj no.
Cia peb xav txog qhov piv txwv teeb meem. 3 x ygo2 - 8x + 4 zoo li nyuaj thaum xub thawj. Txawm li cas los xij, thaum peb paub tias 3 tsuas muaj ob yam tseem ceeb (3 thiab 1), qhov sib npaug no yuav yooj yim dua vim peb paub tias peb cov lus teb yuav tsum yog ntawm daim ntawv (3x +/- _) (x +/- _). Hauv qhov no, ntxiv -2 rau ob qho tib si muab cov lus teb raug. -2 × 3x = -6x thiab -2 × x = -2x. -6x thiab -2x ntxiv rau -8x. -2 × -2 = 4, yog li peb tuaj yeem pom tias cov ntsiab lus ua piv txwv hauv kab lus thaum sib tw tsim cov khoom qub.
Kauj Ruam 4. Kho los ntawm ua kom tiav cov xwm txheej
Hauv qee kis, kev ua zauv sib npaug tuaj yeem ua tau sai thiab yooj yim suav nrog siv cov cim tshwj xeeb algebraic. Txhua qhov sib npaug ntawm plaub fab hauv daim ntawv x2 2 xh + h2 = (x + h)2. Yog li yog tias hauv koj qhov kev ua zauv koj tus nqi b yog ob npaug ntawm lub hauv paus c ntawm koj tus c, koj qhov kev ua zauv tuaj yeem muab sib piv rau (x + (hauv paus (c)))2.
Piv txwv li, kab zauv x2 +6x+9 muaj cov duab no. 32 yog 9 thiab 3 × 2 yog 6. Yog li, peb paub tias daim ntawv ntawm qhov sib npaug no yog (x + 3) (x + 3), lossis (x + 3)2.
Kauj Ruam 5. Siv cov zauv los daws qhov sib npaug sib npaug
Txawm hais tias koj tau txiav txim siab li cas ntawm koj qhov kev ua zauv sib npaug, thaum qhov kev ua zauv yog qhov tseeb, koj tuaj yeem nrhiav cov lus teb tau rau tus nqi x los ntawm kev ua txhua qhov sib npaug rau xoom thiab daws lawv. Txij li thaum koj tab tom nrhiav tus nqi ntawm x uas ua rau koj qhov kev ua zauv sib npaug rau xoom, tus nqi ntawm x uas ua rau txhua yam sib npaug rau xoom yog cov lus teb tau rau koj qhov kev ua zauv sib npaug.
Cia peb rov qab mus rau kab zauv x2 + 5x + 6 = 0. Qhov kev ua zauv no tau muab rau hauv (x + 3) (x + 2) = 0. Yog tias ib qho twg sib npaug 0, txhua qhov sib npaug sib npaug 0, yog li peb cov lus teb tau rau x yog tus lej- tus lej uas ua rau (x + 3) thiab (x + 2) sib npaug 0. Cov lej no yog -3 thiab -2, feem.
Kauj Ruam 6. Txheeb xyuas koj cov lus teb - qee cov lus teb yuav yuam kev
Thaum koj pom cov lus teb tau rau x, ntsaws lawv rov qab rau hauv koj thawj qhov sib npaug kom pom tias cov lus teb raug. Qee zaum, cov lus teb uas koj pom tsis ua qhov sib npaug qub sib npaug rau xoom thaum rov nkag mus. Peb hu cov lus teb no tsis txawv thiab tsis quav ntsej nws.
-
Cia peb muab -2 thiab -3 rau x2 + 5x + 6 = 0. Thawj, -2:
- (-2)2 + 5(-2) + 6 = 0
- 4 + -10 + 6 = 0
- 0 = 0. Cov lus teb no raug, yog li -2 yog cov lus teb raug.
-
Tam sim no, cia peb sim -3:
- (-3)2 + 5(-3) + 6 = 0
- 9 + -15 + 6 = 0
- 0 = 0. Cov lus teb no tseem raug, yog li -3 yog cov lus teb raug.
Txoj Kev 3 ntawm 3: Txheeb Xyuas Lwm Yam Sib Piv
Kauj Ruam 1. Yog hais tias qhov kev ua zauv tau qhia hauv daim ntawv a2-b2, ua rau (a+b) (a-b).
Kev sib npaug nrog ob qhov sib txawv muaj qhov sib txawv ntau dua li qhov sib npaug quadratic yooj yim. Rau qhov sib npaug a2-b2 txhua yam uas a thiab b tsis sib npaug rau 0, cov ntsiab lus ntawm qhov sib npaug yog (a+b) (a-b).
Piv txwv li, qhov sib npaug 9x2 - 4 xyi2 = (3x + 2y) (3x - 2y).
Kauj Ruam 2. Yog hais tias qhov kev ua zauv tau qhia hauv daim ntawv a22ab+b2, ua rau (a+b)2.
Nco ntsoov tias, yog tias trinomial yog ntawm daim ntawv a2-2ab+b2, daim ntawv foos sib txawv me ntsis: (a-b)2.
4 x.z sib2 + 8xy + 4y2 tuaj yeem rov sau dua li 4x2 + (2 × 2 × 2) xy + 4y2. Tam sim no, peb tuaj yeem pom tias daim ntawv raug lawm, yog li peb tuaj yeem paub tseeb tias qhov tseem ceeb ntawm peb qhov sib npaug yog (2x + 2y)2
Kauj Ruam 3. Yog hais tias qhov kev ua zauv tau qhia hauv daim ntawv a3-b3, muab rau hauv (a-b) (a2ab+ab2).
Thaum kawg, nws twb tau hais tias cov lej sib npaug thiab txawm tias muaj zog ntau dua, tuaj yeem suav nrog, txawm hais tias cov txheej txheem txheej txheem sai dhau los ua qhov nyuaj heev.
Npr, 8 x3 - 27 xyoo3 muab tso rau hauv (2x - 3y) (4x2 + ((2x) (3y)) + 9y2)
Lub tswv yim
- a2-b2 tuaj yeem ua qhov tseeb, a2+b2 tsis tuaj yeem ua qhov tseeb.
- Nco ntsoov yuav ua li cas txhawm rau qhov tsis tu ncua. Qhov no yuav pab tau.
- Ceev faj nrog cov feem me me hauv cov txheej txheem ua haujlwm thiab ua haujlwm nrog cov zauv kom raug thiab ua tib zoo.
- Yog tias koj muaj trinomial ntawm daim ntawv x2bx+ (b/2)2, daim ntawv foos yog (x+(b/2))2. (Koj tuaj yeem ntsib qhov xwm txheej no thaum ua tiav lub xwmfab.)
- Nco ntsoov tias a0 = 0 (cov khoom ntiag tug ntawm cov khoom xoom).
