Mehlala ea Lipotso tse Buisanang ka Lintlha le Li-Generator tsa Zero tsa Li-Polynomial
Ho utloisisa lintlha le li-zero tsa polynomial ke motheo oa bohlokoa ho algebra le calculus. Sehloohong sena, re tla buisana ka mokhoa oa ho fumana lintlha le li-zero tsa polynomial ka mehlala le lipuisano tse qaqileng. Re tla akaretsa mehato e amehang ts'ebetsong le ho utloisisa litlamorao tsa lipalo tsa li-zero le lintlha tsena.
Selelekela sa Lintlha le Lijenereithara tsa Zero tsa Polynomials
Polynomial ke polelo ea lipalo e nang le li-variable le li-coefficient tse hokahaneng le ts'ebetso ea ho eketsa, ho ntša le ho atisa. Sebopeho se akaretsang sa polynomial ke:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0 \]
Mabaka a polynomial ke dipolelwana tse ding tsa polynomial tse ka atiswang ho hlahisa polynomial ya pele, ha di-zero (kapa metso) tsa polynomial e le boleng ba x bo etsang \( P(x) = 0 \).
Mohlala oa 1: Ho Fumana Li-Zero tsa Polynomial e Bonolo
Potso: Fumana bo-zero ba polynomial \( P(x) = x^2 – 5x + 6 \).
Puisano:
Ho fumana li-zero tsa polynomial ena, re hloka ho fumana boleng ba x bo etsang hore polynomial e lekane le zero. Ka mantsoe a mang, re hloka ho rarolla equation:
\[ x^2 – 5x + 6 = 0 \]
Mohato oa pele ke ho leka ho lekanya palo ea polinomial. Ha re batle linomoro tse peli tseo sehlahisoa sa tsona e leng +6 le tseo kakaretso ea tsona e leng -5. Linomoro tsena ke -2 le -3. Ka hona, palo ea polinomial e ka hlalosoa e le:
\[(x – 2)(x – 3) = 0 \]
Jwale, re sebedisa molaomotheo wa lefela-sehlahiswa:
\[(x – 2) = 0 \]
kapa
\[(x – 3) = 0 \]
Kahoo, re fumana:
\[x = 2 \]
dan tao
\[x = 3 \]
Ka hona, di-zero tsa polynomial \( P(x) = x^2 – 5x + 6 \) ke \( x = 2 \) le \( x = 3 \).
Mohlala oa 2: Ho Faktoria Li-polynomial tse Rarahaneng Haholo
Potso: Dintlha tsa polynomial ke dife \( P(x) = 2x^3 – 3x^2 – 8x + 12 \)?
Puisano:
Ho lekanya polynomial ena, re ka sebelisa mokhoa oa synthetic factoring kapa theorem ea factor.
Mohato oa pele ke ho leka ho fumana lefela le le leng la polynomial. Re ka leka boleng bo ka bang teng joalo ka x = 1, -1, 2, -2, 3, -3, 4, -4, jj., ho latela theorem ea factor. A re lekeng x = 2:
\[ P(2) = 2(2)^3 – 3(2)^2 – 8(2) + 12 \]
\[ = 2(8) – 3(4) – 16 + 12 \]
\[ = 16 – 12 – 16 + 12 = 0 \]
Kaha P(2) = 0, \( x = 2 \) ke lefela la polynomial. Jwale re ka arola polynomial ka \( x – 2 \) ho fumana dintlha tse ding.
Ho sebelisa karohano ea polynomial:
\[ (2x^3 – 3x^2 – 8x + 12) \div (x – 2) \]
Mehato ke e latelang:
1. Arola 2x^3 ka x ho fumana 2x^2.
2. Atisa 2x^2 ka (x – 2) ho fumana 2x^3 – 4x^2.
3. Tlosa karolo ena ea polynomial ho polynomial ea pele ho fumana x^2 - 8x + 12.
4. Arola x^2 ka x ho fumana x.
5. Atisa x ka (x – 2) ho fumana x^2 – 2x.
6. Tlosa sena ho tse setseng ho fumana -6x + 12.
7. Arola -6x ka x ho fumana -6.
8. Atisa -6 ka (x - 2) ho fumana -6x + 12.
9. Tlosa sena ho karolo e setseng ho fumana karolo e setseng ea 0.
Kahoo, sephetho ho eena ke:
\[ 2x^2 + x – 6 \]
Jwale, re hloka ho lekanya \( 2x^2 + x – 6 \). Ho etsa sena, re fumana dinomoro tse pedi tseo sehlahiswa sa tsona e leng \( 2 \makgetlo -6 = -12 \) mme kakaretso ya tsona e le 1. Dinomoro tsena ke 4 le -3.
Re ka ngola hape polynomial e le:
\[ 2x^2 + 4x – 3x – 6 \]
Ebe re li arola ka lihlopha:
\[ 2x(x + 2) – 3(x + 2) \]
Kahoo, re fumana lintlha tse lokelang ho ba:
\[(x + 2)(2x – 3) \]
Kahoo, lintlha tsa polynomial \( 2x^3 – 3x^2 – 8x + 12 \) ke:
\[(x – 2)(x + 2)(2x – 3) \]
Mohlala oa 3: Jenereithara ea Zero ea Li-polynomial tsa Degree e Phahameng
Potso: Fumana bolefela bohle ba polynomial \( P(x) = x^4 – 4x^3 + 6x^2 – 4x + 4 \).
Puisano:
Ho fumana li-zero tsa polynomial ea boemo bo holimo joalo ka ena, ka linako tse ling re ka sebelisa mekhoa e kang motso o sekoere kapa phetolo, e leng se etsang hore ho be bonolo. A re lekeng ho fumana metso:
Re bona hore polynomial ena e bonahala e le sekwere se phethahetseng sa \( (x - 2) \):
\[(x – 2)^4 = x^4 – 4x^3 + 6x^2 – 4x + 1 \]
Kahoo re fumana ho tsoa ho \( (x – a)^n \) moo n = 4 le a = 2:
Kapa leka ho etsa nolofatso ea motheo le ho bala mofuta oa phetoho ea x moo x = 1 kapa 2 ha re ntse re leka ho nolofatsa palo.
\[ sena se fetoha (x – 1)^3 = x^3 -3x^2 + 3x – 1 \]
Ke phetho, nako le nako ha sephetho se nkeloa sebaka se etsa 0 e setseng kapa che.
Polynomial \( e bolelang ho hlahisa feela (x^2 + x)^n / karolelano e akaretsang ea phetoho e sa fetoheng
Qetello:
Mehlala e kaholimo e bonts'a mehato ea ho fumana lintlha le li-zero tsa li-polynomial tse bonolo le tse rarahaneng haholoanyane. Li-zero tsa li-polynomial li ka fumanoa ka ho lekanya polynomial kapa ho sebelisa mekhoa e meng ea lipalo le ea tlhahlobo. Li-factor tsa polynomial ke litholoana tsa ho arola polynomial hore e be sehlahisoa se otlolohileng le lentsoe la polynomial le nang le tekanyo e tlase. Kutloisiso ena le bokhoni li bohlokoa bakeng sa tlhahlobo e tsoetseng pele ea lipalo le lits'ebetso tse fapaneng tse sebetsang, ho kenyeletsoa fisiks, boenjiniere le moruo.
