Mehlala ea Lipotso le Lipuisano mabapi le Mesebetsi ea Li-polynomial le Li-polynomial
Pendahuluan
Mesebetsi ea polynomial le ea polynomial ke lihlooho tsa bohlokoa lipalo tse hlahang khafetsa lits'ebetsong tse fapaneng tsa saense le boenjiniere. Polynomial ke polelo ea lipalo e nang le li-variable, li-constant, le ts'ebetso ea ho eketsa, ho ntša le ho atisa, 'me e na le li-exponents tse seng tsa negative. Mohlala o bonolo oa polynomial ke \( P(x) = x^2 + 2x + 1 \). Mosebetsi oa polynomial ke mosebetsi o hlalositsoeng ka sebopeho sa polynomial. Sehloohong sena, re tla tšohla mehlala ea mesebetsi ea polynomial le ea polynomial, hammoho le litlhaloso tsa tsona tse qaqileng.
Tlhaloso ea Polynomial
Polynomial ho variable e le 'ngoe \( x \) e ka hlalosoa ka sebopeho se akaretsang ka tsela e latelang:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]
Di mana:
– \( a_n, a_{n-1}, \ldots, a_1, a_0 \) ke di-coefficient tse leng dinomoro tsa nnete.
– \( n \) ke matla a phahameng ka ho fetisisa ao e leng palo e felletseng e seng negative.
Lipotso tsa Mehlala le Puisano
Mohlala oa Potso ea 1: Ho Bala Boleng ba Polynomial
Potso:
Ho fanoe ka polynomial \( P(x) = 3x^3 – 2x^2 + 4x – 5 \). Bala boleng ba \( P(2) \).
Puisano:
Ho bala boleng ba polynomial ho \( x = 2 \), re nkela \( x \) sebaka ka 2 ho polynomial:
\[ P(2) = 3(2)^3 – 2(2)^2 + 4(2) – 5 \]
\[ P(2) = 3 \cdot 8 – 2 \cdot 4 + 4 \cdot 2 – 5 \]
\[ P(2) = 24 – 8 + 8 – 5 \]
\[ P(2) = 19 \]
Kahoo, boleng ba \( P(2) \) ke 19.
Mohlala oa Potso ea 2: Ho Fumana Metso ea Polynomial
Potso:
Fumana metso ea polynomial \( P(x) = x^2 – 5x + 6 \).
Puisano:
Re sebelisa mokhoa oa factorization ho fumana metso ea polynomial:
\[ P(x) = x^2 – 5x + 6 \]
\[ P(x) = (x – 2)(x – 3) \]
Kahoo, metso ke \( x = 2 \) le \( x = 3 \).
Mohlala oa Potso ea 3: Ho Bala Li-derivative tsa Polynomial
Potso:
Ha ho fanoe ka polynomial \( P(x) = 4x^3 – 3x^2 + 2x – 1 \). Bala di-derivative tsa pele le tsa bobedi tsa polynomial.
Puisano:
Setho sa pele sa polynomial \( P(x) \) ke:
\[ P'(x) = \frac{d}{dx}(4x^3 – 3x^2 + 2x – 1) \]
\[ P'(x) = 12x^2 – 6x + 2 \]
Setho sa bobedi sa polynomial \( P(x) \) ke:
\[ P”(x) = \frac{d}{dx}(12x^2 – 6x + 2) \]
\[ P”(x) = 24x – 6 \]
Kahoo, derivative ea pele ea \( P(x) \) ke \( 12x^2 – 6x + 2 \) 'me derivative ea bobeli ke \( 24x – 6 \).
Mohlala Potso ea 4: Ho Fumana Mosebetsi oa Polynomial ho tsoa Lintlheng Tse Fanoeng
Potso:
Fumana mosebetsi oa polynomial oa boemo ba bobeli \( P(x) \) o fetang lintlheng (1, 2), (2, 3), le (3, 14).
Puisano:
Re nka mosebetsi oa polynomial oa boemo ba bobeli oa foromo:
\[ P(x) = selepe^2 + bx + c \]
Ka ho nkela lintlha sebaka ka polynomial:
1) Ho tloha ho (1, 2): \( a(1)^2 + b(1) + c = 2 \) \(\motsu o nepahetseng a + b + c = 2 \)
2) Ho tloha ho (2, 3): \( a(2)^2 + b(2) + c = 3 \) \(\motsu o nepahetseng 4a + 2b + c = 3 \)
3) Ho tloha ho (3, 14): \( a(3)^2 + b(3) + c = 14 \) \(\motsu o nepahetseng 9a + 3b + c = 14 \)
Ka mor'a moo re na le tsamaiso ea li-equation tse otlolohileng:
\[a + b + c = 2 \]
\[4a + 2b + c = 3 \]
\[9a + 3b + c = 14 \]
Re rarolla tsamaiso ena ea li-equation:
1) Tlosa di-equation tsa bobedi le tsa pele:
\[(4a + 2b + c) – (a + b + c) = 3 – 2 \]
\[ 3a + b = 1 \]
2) Tlosa di-equation tsa boraro le tsa bobedi:
\[ (9a + 3b + c) – (4a + 2b + c) = 14 – 3 \]
\[ 5a + b = 11 \]
Re rarolla tsamaiso ea li-equation:
\[ 3a + b = 1 \]
\[ 5a + b = 11 \]
Tlosa di-equation tsa bobedi le tsa pele:
\[ (5a + b) – (3a + b) = 11 – 1 \]
\[ 2a = 10 \]
\[a = 5 \]
Kenya \( a = 5 \) sebakeng sa e 'ngoe ea li-equation:
\[ 3(5) + b = 1 \]
\[ 15 + b = 1 \]
\[b = -14 \]
Kenya \( a = 5 \) le \( b = -14 \) sebakeng sa e 'ngoe ea li-equation tsa pele:
\[ 5 + (-14) + c = 2 \]
\[ -9 + c = 2 \]
\[c = 11 \]
Kahoo, mosebetsi oa polynomial o fetang lintlheng tsena ke:
\[ P(x) = 5x^2 – 14x + 11 \]
Ho koala
Sehloohong sena, re buisane ka mathata a 'maloa a mehlala a amanang le mesebetsi ea polynomial le polynomial, hammoho le mokhoa oa ho a rarolla. Mathata ana a tloha ho bala boleng ba polynomial, ho fumana metso ea polynomial, ho bala derivative ea polynomial, ho ea ho ho fumana mosebetsi oa polynomial ho tsoa lintlheng tse tsejoang. Mesebetsi ea polynomial le polynomial ke motheo oa likhopolo tse ngata tse tsoetseng pele tsa lipalo, joalo ka tlhahlobo ea lipalo, algebra e otlolohileng, le khopolo-taba ea linomoro. Ho utloisisa metheo ena ho bohlokoa bakeng sa katleho masimong a fapaneng a thuto le a profeshenale.
