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

 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