Khopolo ea Li-polynomial le Thepa ea Tsona
Li-polynomial (kapa li-polynomial) ke mohopolo oa motheo lipalo, o sebelisoang haholo ho algebra, calculus, lipalo-palo, le ho etsa mohlala oa liketsahalo tsa lefats'e la nnete tse kang kholo ea baahi, litsela tsa motsamao, le ntlafatso. Ho sa tsotellehe bonolo ba tsona bo bonahalang, li-polynomial li na le sebopeho se hlalositsoeng hantle le thepa ea bohlokoa e nolofalletsang ts'ebetso ea lipalo e hlophisitsoeng. Sengoloa sena se tšohla tlhaloso ea li-polynomial, sebopeho sa tsona se akaretsang, likhato, mefuta, ts'ebetso ea motheo, le thepa ea bohlokoa e hlokahalang ho utloisisoa.
Tlhaloso ea Polynomial
Ka kakaretso, polynomial ke polelo ea algebra e entsoeng ka ho eketsa le/kapa ho ntša mantsoe a 'maloa, ao e 'ngoe le e 'ngoe ea 'ona e leng coefficient e atisitsoeng ke phetoho e phahamisitsoeng ho matla a kakaretso a seng negative. Ka mantsoe a mang, matla a phetoho ho polynomial ha ea lokela ho ba negative 'me ha ea lokela ho ba karoloana.
Mehlala ea lipalo-palo tsa polynomial:
– \( 3x^2 + 2x – 5 \)
– \( x^4 – 7x^2 + 1 \)
– \( 6 \) (mantswe a sa fetoheng le ona ke a di-polynomial)
Hase polynomial:
– \( \frac{2}{x} = 2x^{-1} \) (matla a fosahetseng)
– \( \sqrt{x} = x^{1/2} \) (matla a karoloana)
– \( 3x^2 + \frac{1}{x^3} \) (e na le matla a fosahetseng)
Sebopeho se Akaretsang sa Li-polynomial
Polenomiale ea phetoho e le 'ngoe (mohlala, phetoho \(x\)) e ka ngoloa ka mokhoa o latelang:
\[
P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_2x^2 + a_1x + a_0
\]
ka:
– \( a_n, a_{n-1}, \ldots, a_0 \) ke di-coefficient (dinomoro tsa nnete, tse utlwahalang, kapa tse rarahaneng),
– \( n \) ke palo e felletseng e seng negative,
– \( a_n \neq 0 \) hoo tekanyo ea polynomial e leng \(n\).
Poleloana \(a_n x^n\) e bitsoa lentsoe le etellang pele, 'me \(a_n\) e bitsoa coefficient e etellang pele.
Tekanyo ea Polynomial
Tekanyo ea polynomial ke matla a phahameng ka ho fetisisa a phetoho ho polynomial e nang le coefficient e seng ea lefela.
Contoh:
– \(2x^5 + x^2 – 1 \) o na le lengolo la 5
– \(7x – 3 \) o na le lengolo la 1
– \(9 \) e na le tekanyo ea 0 (polynomial e sa fetoheng)
Tekanyo e fana ka tlhahisoleseding ea bohlokoa, mohlala mabapi le sebopeho sa kerafo, palo e phahameng ka ho fetisisa ea metso, le boitšoaro ba polynomial ha \(x\) e le kholo haholo kapa e nyane haholo.
Mefuta ea Li-polynomial ho latela Palo ea Mantsoe
Li-polynomial li ka boela tsa aroloa ho latela palo ea mantsoe:
1. Monom: lentsoe le le leng, mohlala \( 5x^3 \)
2. Binomial: mantsoe a mabedi, mohlala \( x^2 – 4 \)
3. Trinomial: mantsoe a mararo, mohlala \( x^2 + 2x + 1 \)
4. Polynomial (kakaretso): mantsoe a fetang a mararo, mohlala \( x^4 + x^3 – 2x^2 + 7x – 1 \)
Mesebetsi ea Motheo ho Li-polynomial
1. Ho eketsa le ho tlosa
Ho eketsa/ho ntsha di-polynomial ho etswa ka ho kopanya mantswe a tshwanang (a nang le di-variable le matla a tshwanang).
Contoh:
\[
(2x^2 + 3x – 1) + (x^2 – 5x + 4) = 3x^2 – 2x + 3
\]
2. Ho atisoa
Ho ata ha di-polynomial ho etswa ka ho aba nako e nngwe le e nngwe ho polynomial ya pele hodima nako e nngwe le e nngwe ho polynomial ya bobedi.
Contoh:
\[
(x+2)(x-3) = x^2 -3x + 2x – 6 = x^2 – x – 6
\]
3. Karohano ea Li-polynomial
Karohano ea li-polynomial e tšoana le karohano ea linomoro, hangata e bitsoang karohano e telele kapa e ka sebelisa karohano ea maiketsetso bakeng sa likarohano ka mokhoa oa \(xa\).
Karohano ena e bohlokoa bakeng sa ho fumana lintlha, metso, le ho nolofatsa mesebetsi e utloahalang.
Matlotlo a Bohlokoa a Li-polynomial
1. Tlhaho e Koetsoeng (Ho Koaloa)
Sete ea polynomial e koetsoe tlas'a ho eketsa, ho tlosa le ho atisa. Sena se bolela hore haeba \(P(x)\) le \(Q(x)\) e le polynomial, joale:
– \(P(x) + Q(x)\) ke polynomial,
– \(P(x) – Q(x)\) ke polynomial,
– \(P(x)\cdot Q(x)\) ke polynomial.
Leha ho le jwalo, karohano ha se kamehla e hlahisang polynomial. Mohlala:
\[
\frac{x^2+1}{x+1}
\]
sephetho e ka ba polynomial plus remain, kapa esita le mosebetsi o utloahalang haeba o sa aroloe ke .
2. Sephetho sa Tekanyo ea Ts'ebetso
Haeba \(P(x)\) e na le tekanyo \(m\) mme \(Q(x)\) e na le tekanyo \(n\), joale:
– Tekanyo e phahameng ka ho fetisisa ya \(P(x)+Q(x)\) ke \(\max(m,n)\) (e ka ba nyane haeba dipehelo tse hodimo di hlakolana).
– Degree \(P(x)\cdot Q(x) = m+n\) (ka coefficient e etellang pele e sa faneng ka lefela).
– Karohanong ya \(P(x):Q(x)\), tekanyo ya quotient e ka ba \(mn\) haeba \(m \ge n\).
3. Khopolo-taba ea Lintlha
E 'ngoe ea litšobotsi tsa bohlokoa ka ho fetisisa ke kamano pakeng tsa lintlha le metso. Theorem ea lintlha e re:
\[
(xa) \text{ ke ntlha } P(x) \iff P(a)=0
\]
Ke hore, haeba phetolo \(x=a\) e hlahisa lefela, joale \(xa\) e tlameha ho arola polynomial ka ho lekana.
Mohlala: Haeba \(P(2)=0\), joale \(x-2\) ke ntlha ea \(P(x)\).
4. Khopolo-taba e setseng
Haeba polynomial \(P(x)\) e arotsoe ka \(xa\), joale karolo e setseng ea karohano ke \(P(a)\).
Sena se etsa hore ho be bonolo ho lekola karolo e setseng ntle le ho etsa karohano e telele.
5. Palo ea Metso
Polynomial ea tekanyo \(n\) e na le metso ea 'nete e ikhethang bonyane \(n\). Linomorong tse rarahaneng, polynomial ea tekanyo \(n\) e na le metso e nepahetseng hantle \(n\) (ho nahanoa ka bongata ba metso), ho latela khopolo-taba ea motheo ea algebra.
Contoh:
– Polenomial ea tekanyo ea 2 e na le metso ea sebele e ka bang 2.
– Polenomial ea tekanyo ea 3 e na le metso ea sebele e ka bang 3.
6. Qetella Boitšoaro
Thepa e 'ngoe ea bohlokoa, haholo-holo bakeng sa ho utloisisa li-graph, ke boitšoaro ba polynomial ha \(x \to \infty\) kapa \(x \to -\infty\). Boitšoaro bona bo khethoa ke lentsoe le etellang pele \(a_n x^n\):
– Haeba \(n\) e lekana le \(a_n > 0\), kerafo e ntse e eketseha mahlakoreng ka bobedi.
– Haeba \(n\) e lekana mme \(a_n < 0\), kerafo e theohela tlase mahlakoreng ka bobedi. - Haeba \(n\) e le odd mme \(a_n > 0\), kerafo e wela ka letsohong le letshehadi mme e nyoloha ka ho le letona.
– Haeba \(n\) e le odd le \(a_n < 0\), kerafo e eketseha ka letsohong le letšehali mme e fokotseha ka letsohong le letona. Qetello Polynomial ke polelo ea algebraic e entsoeng ka mantsoe a nang le matla a kakaretso a seng a negative. Mehopolo ea degree, coefficients, le ts'ebetso e etsa hore polynomial e be bonolo ho e sekaseka le ho e sebelisa libakeng tse ngata tsa lipalo le lits'ebetso tsa eona. Thepa ea bohlokoa joalo ka thepa e koetsoeng, molao oa degree, theorem ea factor, theorem e setseng, kakaretso ea metso, le boitšoaro ba ho qetela li fana ka motheo o tiileng oa ho rarolla mathata a algebraic, ho taka li-graph, le ho haha mehlala ea lipalo. Haeba u lakatsa, nka tsoela pele ka mathata a mohlala le lipuisano (mohlala, ho fumana metso ea polynomial, factorization, kapa karohano ea maiketsetso) kapa ho theha mofuta o bonolo oa sengoloa sena bakeng sa baithuti ba sekolo se phahameng/se phahameng.
