Los ntawm kev sau ib daim duab polynomial ua cov sib npaug ntawm lwm cov polynomials, feem ntau peb tuaj yeem qhia qhov kev hais tawm yooj yim.

 Qhov ua kom tshwm sim hauv cov pov thawj

Peb siv hom kev faib tawm no thaum muaj ib yam dab tsi uas rov ua dua nyob rau hauv txhua nqe lus ntawm polynomial.

Qhov kev cuam tshuam no, uas tuaj yeem muaj cov lej thiab ntawv sau, yuav muab tso ua ntej cov chaw sib dhos

Sab hauv cov cim qhwv nws yuav ua tau los ntawm sib faib txhua lub sij hawm ntawm polynomial los ntawm qhov sib txuam.

Hauv kev coj ua, peb yuav ua cov theem nram qab no:

1) Qhia tias yog muaj tus lej twg uas faib rau txhua qhov sib npaug ntawm cov polynomial thiab cov tsiaj ntawv uas pheej hais tas li nyob rau hauv txhua nqe lus.
2) Muab cov khoom sib xws (naj npawb thiab cov tsiaj ntawv) tso rau sab xub ntiag ntawm daim phiaj (hauv cov ntawv pov thawj).
3) Qhov chaw cim daim phiaj rau qhov tshwm sim ntawm faib txhua yam ntawm polynomial los ntawm qhov ntsuas uas yog pov thawj. Hauv cov ntawv sau, peb siv tib txoj cai ntawm kev faib hluav taws xob.

Piv txwv

a) Dab tsi yog cov ntawv txiav txim siab ntawm polynomial 12x + 6y - 9z?

Ua ntej, peb txheeb xyuas tus lej ntawd 3 faib tag nrho cov coefficients thiab tias tsis muaj ntawv sau ua ntxiv.

Peb tso tus naj npawb 3 tso rau ntawm kab thawj coj, peb faib txhua nqe lus los ntawm peb thiab peb yuav tso qhov txiaj ntsig hauv sab hauv kab thaiv:

12x + 6y - 9z = 3 (4x + 2y - 3z)

b) Koom Tes 2a2ib + 3a3c - a4 4.

Vim tias tsis muaj tus lej uas faib 2, 3 thiab 1 nyob rau tib lub sijhawm, peb yuav tsis tso tus lej nyob rau sab xub ntiag ntawm kab caj dab.

Daim ntawv un nws tau rov qab ua rau txhua nqe lus. Qhov tshwm sim ntau yuav yog un2, uas yog qhov tsawg tshaj plaws tshaj plaws kev ntawm un Hauv kev hais tawm.

Peb sib faib txhua nqe lus polynomial los ntawm un2:

2a2 b: ua a2 = 2 a2 - 2 b = 2 ib

3ro3c: aw a2 = 3 a3 - 2 c = 3 aw

un4 4 : a2 = a2

Peb muab lub un2 ua ntej qhov thaiv thiab cov txiaj ntsig ntawm kev faib ua pab hauv kev nkhaus:

2a2ib + 3a3c - a4 4 = a2 (2b + 3ac - a2)

Pawg

Hauv qhov kev txiav txim siab tias tsis muaj qhov rov ua dua hauv txhua nqe lus, peb tuaj yeem siv pawg kev faib tawm.

Rau qhov ntawd, peb yuav tsum txheeb xyuas cov lus siv uas tuaj yeem ua ke los ntawm ntau yam.

Hauv qhov kev txheeb xyuas qhov tseeb no, peb hais txog qhov tshwm sim ntawm cov pawg.

Piv txwv

Qhov txheeb ze polynomial mx + 3nx + my + 3ny

Cov lus mx y 3 nnx muaj raws li ib qho feem cuam tshuam xCov. Cov lus mi y 3ny muaj raws li ib qho feem cuam tshuam y.

Muab cov ntsiab lus no ua pov thawj:

x (m + 3n) + y (m + 3n)

Nco ntsoov tias (m + 3n) tam sim no rov ua ob qho tib si hauv ob lo lus ib yam.

Muab nws rov qab rau hauv cov ntaub ntawv pov thawj, peb pom cov ntaub ntawv pov thawj hauv cov ntawv polynomial:

mx + 3nx + my + 3ny = (m + 3n) (x + y)

Zoo meej square trinomial

Trinomials yog polynomials nrog 3 cov lus.

Zoo meej square trinomials rau2 2ab + b2 thiab cov2 - 2ab + b2 qhov tshwm sim ntawm cov khoom lag luam zoo kawg ntawm hom (a + b)2 thiab (a - b)2.

Yog li ntawd, qhov ntsuas ntawm cov zoo meej square trinomial yuav yog:

un2 2ab + b2 = (a + ib)2 (square ntawm qhov tawm ntawm ob lo lus)

un2 - 2ab + b2 = (a - b)2 (square ntawm qhov sib txawv ntawm ob nqe lus)

Txhawm rau kom paub yog tias qhov trinomial yog qhov zoo nkauj zoo meej, peb ua cov hauv qab no:

1) Xam txog lub hauv paus xwm ceem ntawm cov ntsiab lus uas tshwm hauv square.
2) Tso cov nuj nqis ntawm 2.
3) Sib piv cov nqi pom nrog lub sij hawm uas tsis muaj plaub fab. Yog tias lawv zoo ib yam, nws yog thaj chaw zoo meej.

Piv txwv

a) Xa tau cov polynomial x2 X + 6x + 9

Ua ntej, peb yuav tsum ntsuas seb qhov polynomial yog zoo meej zoo li cas.

X2 = xy √9 = 3

Txauv rau 2, peb pom: 2. 3) x = 6x

Txij li thaum pom tus nqi yog sib npaug rau lub sijhawm tsis-square, polynomial yog ib lub zoo meej square.

Yog li ntawd, qhov ntsuas yuav yog:

x2 X + 6x + 9 = (x + 3)2

b) Koom xwm cov polynomial x2 - 8xy + 9y2

Kev ntsuas zoo meej trinomial square:

X2 = xy √9y2 = 3 xyoos

Ntau tshaj: 2. x. 3y = 6xyooj

Tus nqi pom tsis haum rau lub sijhawm polynomial (8xy ≠ 6xyx).

Vim nws tsis yog lub zoo meej xwm yeem, peb tsis tuaj yeem siv hom kev faib tawm.

Qhov sib txawv ntawm ob fab

Qhov ntsuas polynomials ntawm hom a2 - b2 Peb siv cov khoom lag luam zoo tshaj plaws ntawm qhov sib txawv rau qhov sib txawv.

Yog li ntawd, qhov ntsuas ntawm polynomials ntawm hom no yuav yog:

un2 - b2 = (a + b). (a - b)

Kev tshaj tawm, peb yuav tsum coj lub hauv paus xwm fab ntawm ob nqe lus.

Tom qab ntawv sau qhov khoom ntawm kev suav ntawm qhov tseem ceeb pom thiab qhov sib txawv ntawm cov nqi.

Piv txwv

Factor rau binomial 9x2 - 25.

Ua ntej, nrhiav cov square hauv paus ntawm nqe lus:

9 x ygo2 = 3x thiab √25 = 5

Sau cov nqi no yog cov khoom ntawm qhov tawm thiab qhov sib txawv:

9x2 - 25 = (3x + 5). (3x - 5)

Lub voos xwmfab zoo meej

The polynomials a3 + 3a2b + 3 ab2 + b3 thiab cov3 - Thib 32b + 3 ab2 - b3 qhov tshwm sim ntawm cov khoom lag luam zoo kawg ntawm hom (a + b)3 lossis (a - b)3.

Yog li ntawd, daim ntawv txiav txim siab ntawm lub voos xwmfab zoo meej yog:

un3 + 3a2b + 3 ab2 + b3 = (a + ib)3

un3 - Thib 32b + 3 ab2 - b3 = (a - b)3

Txhawm rau cov ntsiab lus zoo polynomials, peb yuav tsum nqa lub voos xwm txheej ntawm cov nqe lus cubed.

Yog li, koj yuav tsum paub tseeb tias polynomial yog lub voos xwm txheej zoo meej.

Yog tias muaj, peb ntxiv los yog rho tawm qhov tseem ceeb ntawm lub voos cij pom hauv lub voos xwmfab.

Piv txwv

a) Xa tau cov polynomial x3 + 6 x2 X + 12x + 8

Ua ntej, cia peb suav lub hauv paus lub hauv paus ntawm cov lus hauv cubes:

3√ x ib3 = xe 38 = 2

Tom qab ntawv lees tias nws yog lub voos xwmfab zoo meej:

3) xwm2 Cov. 2 = 6x2

3) x. ob2 Luas = 12 x

Txij li cov lus pom tau muaj tib yam li cov lus ntawm polynomial, nws yog lub zoo meej cube.

Yog li ntawd, qhov ntsuas yuav yog:

x3 + 6 x2 X + 12x + 8 = (x + 2)3

b) Koom xwm cov polynomial a3 - cuaj caum2 + 27a - 27

Cia peb xub xam lub hauv paus lub hauv paus ntawm cov lus hauv cubes:

3. A3 = ae 3- 27 = - 3

Txij li cov lus pom tau muaj tib yam li cov lus ntawm polynomial, nws yog lub zoo meej cube.

Yog li ntawd, qhov ntsuas yuav yog:

un3 - cuaj caum2 + 27a - 27 = (a - 3)3

Soled ce

Qhov zoo ntawm cov hauv qab no:

a) 33x + 22y - 55z
b) 6nx - 6ny
c) 4x - 8c + mx - 2mc
d) 49 - a2
e) 9 ib2 Ib + 12a + 4