Ngokubhala i-polynomial njengokuphindaphinda kwezinye iipolynomials, sinokuyenza lula intetho.

Isalathiso semixholo

## Into eqhelekileyo kubungqina

Sisebenzisa olu hlobo lwe-factorization xa kukho into ephindayo kuyo yonke imigaqo ye-polynomial.

Le nto, enokuqulatha amanani kunye neeleta, iya kubekwa phambi kwabazali.

Ngaphakathi kubazali kuya kuba sisiphumo sokwahlula ikota nganye ye-polynomial ngento eqhelekileyo.

Ukuziqhelanisa, siya kwenza la manyathelo alandelayo:

1) Chonga ukuba ngaba kukho inani elihlulahlula zonke izinto ezilandelelanayo ze-polynomial kunye noonobumba abaphindaphindwayo kuwo onke amagama.
2) Beka izinto eziqhelekileyo (inombolo kunye neeleta) phambi kwabazali (kubungqina).
3rd) Beka kubazali phakathi kwesiphumo sokwahlulahlula into nganye ye-polynomial ngento ebubungqina. Kwimeko yoonobumba sisebenzisa umgaqo ofanayo wokwahlulahlula amandla.

### U mzekelo

a) Yeyiphi le fomu ifakelweyo ye-polynomial 12x + 6y-9z?

Kuqala, sichonga ukuba inani 3 Yahlulahlula yonke into eyenziweyo kunye nokuba akukho leta iphindaphindwayo.

Sibeka inamba yesi-3 phambi kwabazali, sahlulahlula yonke imiqathango kathathu kwaye siza kubeka iziphumo ngaphakathi kwabazali.

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

b) Inqaku 2a2b + 3a3c - a4 4.

Kuba akukho nani lahlulahlula u-2, 3 no-1 ngaxeshanye, asizukubeka naliphi na inani phambi kwabazali.

Ileta un iphindaphindwe ngokwemigaqo yonke. Into eqhelekileyo iya kuba un2, lelona lelona zincinci lincinci un Kwintetho.

Sisahlula igama ngalinye le-polynomial ngo un2:

2a2 b: uya2 = 2a2 - 2 b = 2b

I-3ro3c:a2 = 3a3 - 2 c = 3ac

un4 4 : a2 = a2

Sibeka un2 ngaphambi komzali kunye neziphumo ze izahlulo kwizibiyeli:

2a2b + 3a3c - a4 4 = a2 (2b + 3ac - a2)

## Iqela

Kwi-polynomial ukuba akukho nto iphindaphindayo kuyo yonke imigaqo, singasebenzisa ukubeka izinto ngokwamaqela.

Ngenxa yoko, kufuneka sichonge imigaqo enokuthi ihlelwe ngamaqela aqhelekileyo.

Kolu hlobo lokufaka izinto, siqaqambisa izinto eziqhelekileyo zamaqela.

### Umzekelo

Inqaku le-polynomial mx + 3nx + yam + 3ny

Imigaqo mx y 3nx inento eqhelekileyo x. Imigaqo mi y 3ny ndinento eqhelekileyo y.

Ukubeka ezi zinto kubungqina:

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

Qaphela ukuba (m + 3n) ngoku iphinda yomibini imiqathango ngokunjalo.

Ukuyibeka kubungqina, sifumana ifomathi ye-polynomial:

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

## Isikwere esigqibeleleyo se-trinomial

I-Trinomials zii-polynomials ezinamagama ama-3.

Iitrinomials ezigqibeleleyo zesikwere se2 + 2ab + b2 kunye2 - 2ab + b2 iziphumo zemveliso ephawulekayo yohlobo (a + b)2 kunye (a - b)2.

Ke ngoko, ukubekwa kwinqanaba lesikwere esigqibeleleyo se-trinomial kuya kuba:

un2 + 2ab + b2 = (a + b) usetyenziso2 (isikwere sesixa semigaqo emibini)

un2 - 2ab + b2 = (a - b)2 (isikwere somahluko wamagama amabini)

Ukufumanisa ukuba i-trinomial iyisikwere esifanelekileyo, senza oku kulandelayo:

1) Bala ingcambu yesikwere yemigaqo evela kwisikwere.
2) Phindaphinda amaxabiso afunyenwe ngo-2.
3) Thelekisa ixabiso elifunyenweyo kunye nekota engenazo izikwere. Ukuba bayafana, sisikwere esifanelekileyo.

### U mzekelo

a) Ubungakanani bepolynomial x2 + 6x + 9

Kuqala, kufuneka sivavanye ukuba i-polynomial isikwere esifanelekileyo.

√x2 = xy -9 = 3

Ukuphinda-phinda ngo-2, sifumana: 2. 3) x = 6x

Kuba ixabiso elifunyenweyo lilingana nekota engenasikwere, ipolynomial sisikwere esifanelekileyo

Ke ngoko, umba wokulinganisa uya kuba:

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

b) Ubungakanani bepolynomial x2 - 8xy + 9y2

Uvavanyo olugqibeleleyo lwesikwere kathathu:

√x2 = xy -9y2 = 3y

Yandisa: 2. x. 3y = 6xy

Ixabiso elifunyenweyo alihambelani nexesha le-polynomial (8xy ≠ 6xy).

Kuba ayisosikwere esigqibeleleyo se-trinomial, asinakulusebenzisa olu hlobo lwezinto.

## Umahluko wezikwere ezibini

Ukwenza iipolynomials zohlobo a2 - b2 Sisebenzisa imveliso ephawulekayo yesixa-mali umahluko.

Ke ngoko, ubungakanani beepolynomials zolu hlobo ziya kuba:

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

Ukulinganisa, kufuneka sithathe ingcambu yesikwere yala magama mabini.

Emva koko bhala imveliso yamaxabiso amaxabiso afunyenweyo numahluko wamaxabiso lawo.

### Umzekelo

Inqaku le-binomial 9x2 - 25.

Okokuqala, fumana ingcambu yesikwere semigaqo:

X9x2 = 3x kunye no -25 = 5

Bhala la maxabiso njengemveliso yesixa kunye nolwahluko:

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

## Ityhubhu egqibeleleyo

Iipolynomials a3 + 3a2b + 3ab2 + b3 kunye3 Okwesithathu2b + 3ab2 - b3 iziphumo zemveliso ephawulekayo yohlobo (a + b)3 okanye (a - b)3.

Ke ngoko, ifomathi efakwe kwikhabhathi efanelekileyo yile:

un3 + 3a2b + 3ab2 + b3 = (a + b) usetyenziso3

un3 Okwesithathu2b + 3ab2 - b3 = (a - b)3

Ukwenza i-polynomials ezinjalo, kufuneka sithathe ingcambu yecube yamagama e-cubed.

Ke, kuya kufuneka uqinisekise ukuba i-polynomial yityhubhu efanelekileyo.

Ukuba kunjalo, sidibanisa okanye sithabathe amaxabiso eengcambu zetyhubhu ezifumaneka kwityhubhu.

### U mzekelo

a) Ubungakanani bepolynomial x3 + 6x2 + 12x + 8

Okokuqala, masibe ingcambu yecube yamacandelo kwiityhubhu:

3X3 = xe 38 = 2

Emva koko qinisekisa ukuba yityhubhu efanelekileyo:

3) x2 . 2 = 6x2

3) x. Mbini2 = 12x

Kuba imigaqo efunyenweyo iyafana nemigaqo ye-polynomial, yityhubhu efanelekileyo.

Ke ngoko, umba wokulinganisa uya kuba:

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

b) Ubungakanani bepolynomial a3 amashumi alithoba2 + 27a - 27

Masiqale sibale ingcambu yetyhubhu yamagama kwiityhubhu:

3. A3 = ae 3√ - 27 = - 3

Kuba imigaqo efunyenweyo iyafana nemigaqo ye-polynomial, yityhubhu efanelekileyo.

Ke ngoko, umba wokulinganisa uya kuba:

un3 amashumi alithoba2 + 27a - 27 = (a - 3)3

## Isonjululwe imithambo

I-factor polynomials elandelayo:

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