Nā Laʻana o nā Nīnau a me nā Kūkākūkā e pili ana i nā Polynomials a me nā Hana Polynomial
Pendahuluan
He mau kumuhana koʻikoʻi nā polynomials a me nā hana polynomial i ka makemakika e ʻike pinepine ʻia ana i nā noi ʻepekema a me nā ʻenekinia like ʻole. ʻO ka polynomial kahi hōʻike makemakika i haku ʻia me nā loli, nā mea mau, a me nā hana o ka hoʻohui, ka hoʻemi ʻana, a me ka hoʻonui ʻana, a he mau exponents ʻaʻole maikaʻi ʻole. ʻO kahi laʻana maʻalahi o ka polynomial ʻo \( P(x) = x^2 + 2x + 1 \). ʻO ka hana polynomial kahi hana i hōʻike ʻia ma ke ʻano polynomial. Ma kēia ʻatikala, e kūkākūkā mākou i nā laʻana o nā polynomials a me nā hana polynomial, me kā lākou wehewehe kikoʻī.
Wehewehena o ka Polynomial
Hiki ke hōʻike ʻia kahi polynomial i loko o hoʻokahi loli \( x \) ma ke ʻano laulā penei:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]
Ma hea:
– ʻO \( a_n, a_{n-1}, \ldots, a_1, a_0 \) he mau coefficients he mau helu maoli.
– ʻO \( n \) ka mana kiʻekiʻe loa he helu helu ʻaʻole maikaʻi ʻole.
Nā Nīnau Laʻana a me ke Kūkākūkā
Laʻana Nīnau 1: Ke Helu ʻana i ka Waiwai o kahi Polynomial
Nīnau:
Hāʻawi ʻia kahi polynomial \( P(x) = 3x^3 – 2x^2 + 4x – 5 \). E helu i ka waiwai o \( P(2) \).
Kūkākūkā:
No ka helu ʻana i ka waiwai o ka polynomial ma \( x = 2 \), hoʻololi mākou iā \( x \) me 2 i loko o ka polynomial:
P(2) = 3(2)^3 – 2(2)^2 + 4(2) – 5
\[ P(2) = 3 \cdot 8 – 2 \cdot 4 + 4 \cdot 2 – 5 \]
\[ P(2) = 24 – 8 + 8 – 5 \]
\[ P(2) = 19 \]
No laila, ʻo ka waiwai o \( P(2) \) he 19.
Laʻana Nīnau 2: Ke ʻimi nei i nā aʻa o kahi Polynomial
Nīnau:
E huli i nā aʻa o ka polynomial \( P(x) = x^2 – 5x + 6 \).
Kūkākūkā:
Hoʻohana mākou i ke ʻano factorization e ʻike ai i nā aʻa o ka polynomial:
\[ P(x) = x^2 – 5x + 6 \]
P(x) = (x – 2)(x – 3)
No laila, ʻo nā aʻa \( x = 2 \) a me \( x = 3 \).
Laʻana Nīnau 3: Ke Helu ʻana i nā Polynomial Derivatives
Nīnau:
Hāʻawi ʻia ka polynomial \( P(x) = 4x^3 – 3x^2 + 2x – 1 \). E helu i nā derivatives mua a me ka lua o ka polynomial.
Kūkākūkā:
ʻO ka derivative mua o ka polynomial \( P(x) \) penei:
P'(x) = \frac{d}{dx}(4x^3 – 3x^2 + 2x – 1)
\[ P'(x) = 12x^2 – 6x + 2 \]
ʻO ka lua o ka derivative o ka polynomial \( P(x) \) penei:
\[ P”(x) = \frac{d}{dx}(12x^2 – 6x + 2) \]
\[ P”(x) = 24x – 6 \]
No laila, ʻo ka derivative mua o \( P(x) \) ʻo \( 12x^2 – 6x + 2 \) a ʻo ka derivative ʻelua ʻo \( 24x – 6 \).
Laʻana Nīnau 4: Ke ʻimi nei i kahi Hana Polynomial mai nā Kiko i Hāʻawi ʻia
Nīnau:
E huli i ka hana polynomial kekelē ʻelua \( P(x) \) e hele ana ma waena o nā kiko (1, 2), (2, 3), a me (3, 14).
Kūkākūkā:
Ke manaʻo nei mākou he hana polynomial kekelē ʻelua o ke ʻano:
P(x) = ax^2 + bx + c
Ma ke pani ʻana o nā kiko i loko o ka polynomial:
1) Mai (1, 2): \( a(1)^2 + b(1) + c = 2 \) \(\rightarrow a + b + c = 2 \)
2) Mai (2, 3): \( a(2)^2 + b(2) + c = 3 \) \(\rightarrow 4a + 2b + c = 3 \)
3) Mai (3, 14): \( a(3)^2 + b(3) + c = 14 \) \(\rightarrow 9a + 3b + c = 14 \)
A laila, loaʻa iā mākou kahi ʻōnaehana o nā kaulike linear:
\[ a + b + c = 2 \]
\[ 4a + 2b + c = 3 \]
\[ 9a + 3b + c = 14 \]
Hoʻoponopono mākou i kēia ʻōnaehana o nā hoʻohālikelike:
1) E unuhi i nā kaulike ʻelua a me ka mua:
\[ (4a + 2b + c) – (a + b + c) = 3 – 2 \]
\[ 3a + b = 1 \]
2) E unuhi i nā kaulike ʻekolu a me ka lua:
\[ (9a + 3b + c) – (4a + 2b + c) = 14 – 3 \]
\[ 5a + b = 11 \]
Hoʻoponopono mākou i ka ʻōnaehana o nā hoʻohālikelike:
\[ 3a + b = 1 \]
\[ 5a + b = 11 \]
E unuhi i nā kaulike ʻelua a me ka mua:
\[ (5a + b) – (3a + b) = 11 – 1 \]
\[ 2a = 10 \]
\[ a = 5 \]
E hoʻololi iā \( a = 5 \) i loko o kekahi o nā kaulike:
\[ 3(5) + b = 1 \]
\[ 15 + b = 1 \]
\[ b = -14 \]
E hoʻololi iā \( a = 5 \) a me \( b = -14 \) i loko o kekahi o nā kaulike mua:
\[ 5 + (-14) + c = 2 \]
\[ -9 + c = 2 \]
\[ c = 11 \]
No laila, ʻo ka hana polynomial e hele ana ma o kēia mau kiko:
P(x) = 5x^2 – 14x + 11
Pani
Ma kēia ʻatikala, ua kūkākūkā mākou i kekahi mau pilikia hoʻohālike e pili ana i nā polynomials a me nā hana polynomial, a me pehea e hoʻoponopono ai iā lākou. Loaʻa kēia mau pilikia mai ka helu ʻana i ka waiwai o kahi polynomial, ka loaʻa ʻana o nā aʻa o kahi polynomial, ka helu ʻana i ka derivative o kahi polynomial, a hiki i ka loaʻa ʻana o kahi hana polynomial mai nā kiko i ʻike ʻia. ʻO nā polynomials a me nā hana polynomial ke kumu no nā manaʻo makemakika holomua he nui, e like me ka loiloi helu, ka algebra linear, a me ke kumumanaʻo helu. He mea nui ka hoʻomaopopo ʻana i kēia mau kumu no ka holomua ma nā ʻano ʻoihana like ʻole.
