Imikhiqizo ephawulekayo izinkulumo zezibalo ezivame ukuvela ezimeni ezihlukahlukene futhi zibalulekile ekwenzeni izibalo zibe lula nokuxazulula izinkinga. Kulo mongo, ukuqonda nokuba nolwazi ngemikhiqizo ephawulekayo kubalulekile ocwaningweni lwe-algebra nezibalo ngokuvamile. Kulesi sihloko, sizochaza umqondo wemikhiqizo ephawulekayo, sethule izibonelo ezibalulekile, futhi siphakamise izivivinyo ezixazululiwe ukuze sikusize ubambe futhi uqonde lesi sihloko esibalulekile.
Ukwenza lula incazelo yemikhiqizo emangalisayo ngezinyathelo ezilula nezisebenzayo.
Imikhiqizo ephawulekayo izinkulumo zezibalo ezinefomu elithile, eliphindaphindayo, izibalo ezilula kanye nezibalo ezilula. Ukuze siwuqonde kangcono lo mqondo, ake siwuhlukanise ube izinyathelo ezilula, ezisebenzayo.
Okokuqala, kubalulekile ukuqonda ukuthi imikhiqizo ephawulekayo yakhiwe izinkulumo ze-algebraic ezilandela iphethini echazwe ngaphambilini. Imikhiqizo esemqoka ephawulekayo yilena: isikwele sesamba, isikwele somehluko, umkhiqizo wesamba kanye nomehluko e isikwele se-binomial.
Ukuze ubale le mikhiqizo emangalisayo, mane usebenzise izici zezibalo ezihambisanayo esimweni ngasinye. Ngokwesibonelo, endabeni ye isikwele sesamba, sisebenzisa ifomula (a + b)² = a² + 2ab + b². Kwe isikwele somehluko, sine (a – b)² = a² – 2ab + b².
Ukwenza kube lula ukukuqonda, masixazulule umsebenzi ongokoqobo: bala isikwele sesamba phakathi kuka-3x no-2y. Ukusebenzisa ifomula (a + b)², sine (3x + 2y)² = (3x)² + 2(3x)(2y) + (2y)².
Ngokwenza inkulumo ibe lula, sithola: 9x² + 12xy + 4y². Ngale ndlela, sithola umkhiqizo omangalisayo ohambelana nesikwele sesamba esingu-3x no-2y.
Ngamafuphi, imikhiqizo ephawulekayo izinkulumo zezibalo ezinamafomu ajwayelekile asiza ukubala nokwenza lula izibalo. Ngokuzijwayeza nolwazi lwamafomula afanelekile, kuyenzeka ukuxazulula izinkinga kalula nangokunemba.
Amathiphu okuxazulula izinkinga zomkhiqizo ophawulekayo ngempumelelo nangokoqobo.
Ukuxazulula izinkinga ezibandakanya imikhiqizo ephawulekayo kungaba inselele kubafundi abaningi, kodwa ngamathiphu alungile, kungenzeka ukwenza le nqubo ibe lula futhi isebenze kangcono. Nawa amanye amathiphu okuxazulula izinkinga zomkhiqizo ophawulekayo ngokuphumelelayo nangokoqobo:
1. Khomba uhlobo lomkhiqizo ophawulekayo: Ngaphambi kokuthi uqale ukuxazulula inkinga, khomba ukuthi ingabe isikwele sesamba, isikwele somehluko, umkhiqizo wesamba nomahluko, noma isikwele se-binomial. Ukwazi uhlobo lomkhiqizo kuzokuholela esixazululweni esifanele.
2. Sebenzisa amafomula athile: Uhlobo ngalunye lomkhiqizo ophawulekayo lunefomula ethile yesixazululo sawo. Qiniseka ukuthi uyazazi futhi uzisebenzisa ngendlela efanele enkingeni obhekene nayo.
3. Yenza izinkulumo zibe lula: Izinkinga ezibandakanya imikhiqizo ephawulekayo ngokuvamile zingabonakala ziyinkimbinkimbi ekuqaleni. Ngakho-ke, kubalulekile ukwenza izisho zibe lula futhi ukhombe amaphethini asiza ukuxazulula.
4. Zilolonge ngokuzivocavoca okuhlukahlukene: Ukuzijwayeza kubalulekile ukuze uthole ulwazi ngemikhiqizo emangalisayo. Xazulula izivivinyo ezihlukahlukene, ukushintsha izinhlobo zezinkinga nobunzima, ukuze ucije amakhono akho nokuqonda isihloko.
5. Bheka izinto ezisekelayo: Uma unemibuzo noma ubunzima bokuxazulula inkinga yomkhiqizo, thintana nezincwadi zokufunda, amavidiyo achazayo, noma abafundisi ukuze uthole usizo nokucaciselwa.
Manje njengoba usuwazi amanye amathiphu okuxazulula izinkinga zomkhiqizo ophawulekayo ngempumelelo nangokoqobo, wasebenzise futhi uqinise amakhono akho ezibalo. Ngokuzinikela nangokuphikelela, uzokwazi ukufunda lokhu okuqukethwe futhi uphumelele ezifundweni zakho.
Ukuxazulula imikhiqizo ephawulekayo: umhlahlandlela olula wesinyathelo ngesinyathelo ekuxazululeni lezi zinkulumo ezikhethekile zezibalo.
Imikhiqizo ephawulekayo izinkulumo ezikhethekile zezibalo ezisiza isixazululo sezibalo nokwenza lula ama-polynomials. Ukuze uxazulule imikhiqizo emangalisayo, kubalulekile ukuqonda amafomula futhi uwasebenzise ngendlela efanele. Kulesi sihloko, sizochaza kalula futhi ngokucacile indlela yokuxazulula lezi zinkulumo ezikhethekile zezibalo.
Omunye wemikhiqizo ephawuleka kakhulu iyisikwele sesamba samagama amabili, esingamelwa yifomula: (a + b)² = a² + 2ab + b². Ukuze uxazulule lesi sisho, vele ufake amavelu esikhundleni salokho a e b kufomula futhi wenze imisebenzi yezibalo edingekayo.
Esinye isibonelo somkhiqizo ophawulekayo isikwele somehluko wamagama amabili, esilandela ifomula: (a – b)² = a² – 2ab + b². Ukuze uxazulule lesi sisho, vele ufake amavelu esikhundleni salokho a e b kufomula futhi wenze imisebenzi yezibalo ehambisanayo.
Ngaphezu kwalokhu, kuneminye imikhiqizo ephawulekayo engaba usizo ekuxazululeni izinkinga zezibalo eziyinkimbinkimbi. Kubalulekile ukuzijwayeza ukuxazulula izivivinyo ukuze uzijwayeze lawa mafomula futhi uqinisekise ukusebenza kahle ezivivinyweni nasezivivinyweni zokungena.
Manje njengoba usuqonda ukuthi ungayixazulula kanjani imikhiqizo emangalisayo, zijwayeze ukuxazulula izivivinyo ezilandelayo:
1) Bala inani (3 + 4)²
2) Yenza lula inkulumo (5 – 2)²
Ngalezi zibonelo nokuzijwayeza njalo, uzokwazi ukuxazulula noma yimuphi umkhiqizo ophawulekayo kalula. Khumbula ukubukeza amafomula futhi uzijwayeze njalo ukuze ugcine amakhono akho ezibalo ebukhali!
Zitholele izinhlobo ezintathu zemikhiqizo emangalisayo ngencazelo eyodwa nje elula neqondile.
Imikhiqizo ephawulekayo izinkulumo zezibalo ezinezici ezikhethekile futhi ezingenziwa kalula. Kunezinhlobo ezintathu eziyinhloko zemikhiqizo ephawulekayo: isikwele sesamba, isikwele somehluko e umkhiqizo wesamba kanye nomehluko.
Imikhiqizo ephawulekayo: incazelo kanye nokuzivocavoca okuxazululiwe
Imikhiqizo Okuphawulekayo yimisebenzi ye-algebraic, lapho ukuphindaphinda kwama-polynomials kuvezwa khona, okungadingi ukuxazululwa ngokwesiko, kodwa ngosizo lwemithetho ethile ungathola imiphumela yabo.
Ama-Polynomials aphindaphindwa uma, ngakho-ke, angaba nenani elikhulu lamagama neziguquguqukayo. Ukuze kufinyezwe inqubo, kusetshenziswa imithetho ephawulekayo yomkhiqizo, evumela ukuphindaphinda kwenziwe ngaphandle kokuthi kudlule isikhathi nethemu.
Imikhiqizo ephawulekayo nezibonelo
Umkhiqizo ngamunye ophawulekayo uyifomula ewumphumela we-factorization, owenziwe ngama-polynomials wamagama amaningana, njengama-binomials noma ama-trinomials, abizwa ngokuthi izici.
Izinto ziyisisekelo samandla futhi zine-eksponenti. Uma izici ziphindaphindwa, ama-eksponenti kufanele engezwe.
Kunamafomula omkhiqizo amaningana aphawulekayo, amanye asetshenziswa kakhulu kunamanye, kuye ngama-polynomials, futhi ami kanje:
I-binomial eyisikwele
Ukuphindaphinda kwe-binomial ngokwayo, evezwa ngendlela yamandla, lapho amagama engezwa noma esuswa:
a. Isamba se-binomial sezikwele: ilingana nesikwele sethemu yokuqala, kanye nomkhiqizo wamagama kabili, kanye nesikwele sethemu yesibili. Kuvezwa kanje:
(a+b) 2 =(a+b) * (a + b).
Isibalo esilandelayo sibonisa ukuthi umkhiqizo ukhula kanjani ngokomthetho oshiwo ngenhla. Umphumela ubizwa ngokuthi i-square trinomial ephelele.
Isibonelo 1
(x + 5)² = x² + 2 (x * 5) + 5²
(x + 5) ² = x² + 2 (5x) + 25
(x + 5)² = x² + 10x + 25.
Isibonelo 2
(4a + 2b) = (4a) 2 + 2 (4 * 2b) + (2b) 2
(4a + 2b) = 8a 2 + 2 (8ab) + 4b 2
(4a + 2b) = 8a 2 + 16 ab + 4b 2 .
b. I-Binomial yokukhipha okuyisikwele: umthetho ofanayo wesamba se-binomial uyasebenza, kuphela kulesi simo ithemu yesibili inegethivu. Ifomula yayo imi kanje:
(a - b) 2 = [(a) + (- b)] 2
(a - b) 2 = a 2 + 2 * (-b) + (-b) 2
(a - b) 2 = a 2 - 2ab + b 2 .
Isibonelo 1
(2x–6) 2 =(2x) 2 - 2 (2x * 6) + 6 2
(2x–6) 2 = 4x 2 - 2 (12x) + 36
(2x–6) 2 = 4x 2 24x + 36.
Umkhiqizo we-conjugate binomials
Ama-binomials amabili ahlanganayo lapho amatemu esibili ngayinye enezimpawu ezihlukene, okungukuthi, eyokuqala iphozithivu neyesibili inegethivu, noma ngokuphambene nalokho. Lokhu kuxazululwa ngokusika nokukhipha i-monomial ngayinye. Ifomula imi kanje:
(a+b) * (a - b)
Emfanekisweni olandelayo, umkhiqizo we-conjugate binomials emibili uyathuthukiswa, lapho kungabonakala khona ukuthi umphumela umehluko wezikwele.
Isibonelo 1
(2a + 3b) (2a – 3b) = 4a 2 + (-6ab) + (6ab) + (-9b 2 )
(2a + 3b) (2a – 3b) = 4a 2 - 9b 2 .
Umkhiqizo wama-binomial amabili anegama elivamile
Ingenye yemikhiqizo eyinkimbinkimbi kakhulu futhi engavamile ukusetshenziswa ephawulekayo ngoba iwukuphindaphinda kwama-binomial amabili anetemu elivamile. Umthetho uthi:
- Isikwele setemu elivamile.
- Futhi, engeza amagama angavamile bese uwaphindaphinda ngegama elivamile.
- Kanye nesamba sokuphindaphindwa kwamagama angajwayelekile.
Imelwe kufomula: (x + a) * (x + b) futhi iyanwetshwa njengoba kuboniswe esithombeni. Umphumela uba i-trinomial yesikwele esingaphelele.
(x+6) * (x + 9) = x 2 + (6 + 9) * x + (6 * 9)
(x+6) * (x + 9) = x 2 +15x+54.
Kungenzeka ukuthi igama lesibili (itemu elihlukile) linegethivu futhi ifomula yalo imi kanje: (x + a) * (x – b).
Isibonelo 2
(7x+4) * (7x – 2) = (7x * 7x) + ( 4-2 ) * 7x + (4 * -2)
(7x+4) * (7x – 2) = 49x 2 + (2) * 7x8
(7x+4) * (7x – 2) = 49x 2 + 14x-8.
Kungenzeka futhi ukuthi womabili la magama anegethivu. Ifomula yakho izoba: (x – a) * (x – b).
Isibonelo 3
(3b – 6) * (3b – 5) = (3b * 3b) + (-6-5) * (3b) + (-6 * -5)
(3b – 6) * (3b – 5) = 9b 2 + (-11) * (3b) + (30)
(3b – 6) * (3b – 5) = 9b 2 - 33b + 30.
I-polynomial eyisikwele
Kulokhu, kunamatemu angaphezu kwamabili futhi, ukuyithuthukisa, ngalinye liyisikwele futhi lengezwe ekuphindaphindeni kabili kwethemu elilodwa kwelinye; Ifomula yayo ithi: (a + b + c) 2 futhi umphumela wokusebenza uyi-square trinomial.
Isibonelo 1
(3x + 2y + 4z) 2 =(3x) 2 + (iminyaka engu-2) 2 + (4z) 2 + 2 (6xy + 12xz + 8yz)
(3x + 2y + 4z) 2 = 9x 2 + 4 iminyaka 2 +16z 2 + 12xy + 24xz + 16yz.
I-Binomial ku-cube
Kungumkhiqizo oyinkimbinkimbi omangalisayo. Ukuze uyithuthukise, phindaphinda i-binomial ngesikwele sayo, kanje:
a. Nge-binomial ku-cube yesamba:
- Ikhyubhu yethemu yokuqala, kanye nesikwele esiphindwe kathathu sethemu yokuqala ngokuphindwe kwesibili.
- Kanye nethemu yokuqala kathathu, okwesibili isikwele.
- Kanye ne-cube yethemu yesibili.
(a+b) 3 =(a+b) * (a+b) 2
(a+b) 3 =(a+b) * (a 2 +2ab+b 2 )
(a+b) 3 = a 3 + 2 2 b+ab 2 +ba 2 +2ab 2 + b 3
(a+b) 3 = a 3 + 3 2 b +3ab 2 + b 3 .
Isibonelo 1
(a +3) 3 = a 3 + 3 (a) 2 * (3) + 3 (a) * (3) 2 + (3) 3
(a +3) 3 = a 3 + 3 (a) 2 * (3) + 3 (a) * (9) + 27
(a +3) 3 = a 3 + 9 kube 2 + 27a + 27.
b. Ku-binomial ku-cube yokukhipha:
- Ikhyubhu yethemu yokuqala, susa ngokuphindwe kathathu kwesikwele sethemu yokuqala ngokuphindwe kwesibili.
- Kanye nethemu yokuqala kathathu, okwesibili isikwele.
- Khipha ikhyubhu yethemu yesibili.
(a - b) 3 = (a - b) * (a - b) 2
(a - b) 3 = (a - b) * (a 2 - 2ab + b 2 )
(a - b) 3 = a 3 - 2a 2 b+ab 2 -ba 2 +2ab 2 - b 3
(a - b) 3 = a 3 - 3a 2 b +3ab 2 - b 3 .
Isibonelo 2
(b – 5) 3 =b 3 + 3 (b) 2 * (-5) + 3 (b) * (-5) 2 + (-5) 3
(b – 5) 3 =b 3 + 3 (b) 2 * (-5) + 3 (b) * (amashumi amabili nambili
(b – 5) 3 =b 3 - 15b 2 + 75b – 125.
I-cube ye-trinomial
Iphindwe ngesikwele sayo. Kungumkhiqizo obanzi kakhulu, ngoba kukhona amatemu amathathu aphindwe kabili, kanye nokuphindwe kathathu ithemu ngayinye isikwele, kuphindwe ngokwetemu ngayinye, kuhlanganiswe nomkhiqizo ophindwe kasithupha umkhiqizo wamatemu amathathu. Indlela engcono yokuyibuka yile:
(a+b+c) 3 = (a+b+c) * (a+b+c) 2
(a+b+c) 3 = (a+b+c) * (a 2 + b 2 +c 2 + 2ab + 2ac + 2bc)
(a+b+c) 3 = a 3 + b 3 +c 3 + 3 2 b +3ab 2 + 3 2 c +3c 2 +3b 2 c+3bc 2 + 6abc.
Isibonelo 1
Ukuzivocavoca okuxazululiwe emikhiqizweni ephawulekayo
Ukuzivocavoca 1
Yakha i-binomial elandelayo ye-cube: (4x - 6) 3 .
Isixazululo
Ukukhumbula ukuthi i-binomial ye-cube ilingana ne-cubed yethemu yokuqala, susa ngokuphindwe kathathu isikwele sethemu yokuqala ngeyesibili; kuhlanganisa kathathu ithemu yokuqala, okwesibili isikwele, kukhishwe ikhyubhu yethemu yesibili.
(4x–6) 3 =(4x) 3 - 3 (4x) 2 (6) + 3 (4x) * (6) 2 - (i-6) 2
(4x–6) 3 = 64x 3 - 3 (16x 2 ) (6) + 3 (4x) * (36) - 36
(4x–6) 3 = 64x 3 - 288x 2 + 432x-36.
Ukuzivocavoca 2
Yakha i-binomial elandelayo: (x + 3) (x + 8).
Isixazululo
Kukhona i-binomial lapho kukhona khona igama elivamile, elingu-x, kanti igama lesibili lithi positive. Ukuze uyithuthukise, vele usikwele igama elivamile, kanye nesamba samagama angajwayelekile (3 kanye no-8), bese uwaphindaphinda ngegama elivamile, kanye nesamba sokuphindaphinda kwamagama angavamile.
(x + 3) (x + 8) = x 2 + (3 + 8) x + (3 * 8)
(x + 3) (x + 8) = x 2 +11x+24.
Izinkomba
- Angel, AR (2007). I-Algebra yokuqala Imfundo e-Pearson.
- U-Arthur Goodman, L.H. (1996). I-Algebra ne-trigonometry ene-analytic geometry. Imfundo yePearson.
- Das, S. (n.d.). Izibalo Plus 8. E-United Kingdom: Ratna Sagar.
- Jerome E. Kaufmann, K. L. (2011). I-Algebra Eyisisekelo Nemaphakathi: Indlela Ehlanganisiwe. EFlorida: Isifundo Sokufunda.
- Pérez, C. D. (2010). Imfundo yePearson.