Ngā Tauwehe me ngā Kore o ngā Poronomia
He ariā nui ngā pūrau i roto i te pāngarau, e kitea pinepine ana i roto i ngā momo mara o te pūtaiao me te hangarau. I tōna āhua whānui, ko te pūrau he kīanga taurangi e tito ana i ngā kupu i hangaia e ngā taurangi, ngā tauwehenga, me ngā taupū o ngā taurangi kua whakanuia ki ngā tauoti kore-kino. I roto i tēnei tuhinga, ka matapakihia e mātou ngā ariā nui e rua e hono pinepine ana ki ngā pūrau: ngā tauwehenga me ngā whakaputa kore.
Te Whakamāramatanga o te Poronomiera
I mua i te ruku hohonu atu ki ngā tauwehe me ngā whakaputa kore, me arotake tātou he aha te poronomial. Ka taea te tuhi i te poronomial i roto i te taurangi kotahi x i te āhua whānui penei:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0 \]
Kei hea:
– Ko \( a_n, a_{n-1}, …, a_1, a_0 \) ngā tauwehenga o te pūrinōmiera me \( a_n \neq 0 \).
– Ko te \( n \) te tohu o te pūrau, arā, ko te mana teitei rawa atu o te taurangi \( x \).
Ko tētahi tauira māmā o te pūrau ko \( P(x) = 2x^3 – 3x^2 + x – 5 \).
Ngā Tauwehe Pūrau
Ko ngā tauwehe o tētahi poronomial he poronomial kē atu, ina whakareatia ngātahi, ka puta te poronomial taketake. Hei tauira, ka taea te whakarea i te poronomial \( P(x) = x^2 – 5x + 6 \) ki roto \( (x – 2)(x – 3) \). Ki te whakareatia ēnei poronomial e rua, ka puta te poronomial taketake:
\[ (x – 2)(x – 3) = x^2 – 3x – 2x + 6 = x^2 – 5x + 6 \]
Ko ngā pūrinōmia ((x – 2)) me ((x – 3)) he tauwehe o te pūrinōmia (P(x)).
Tikanga Whakawehewehe
He maha ngā tikanga mō te tauwehe i ngā pūrinōmia, ko ētahi o ēnei:
1. Te Whakawehewehenga me te Whakawehewehenga Taketake:
Ka whakamahia tēnei tikanga hei tauwehe i ngā pūrinōmia he āhua tapawhā, he āhua māmā rānei. Hei tauira, ka taea te tauwehe i te \( x^2 – x – 12 \) ki roto i te \( (x – 4)(x + 3) \).
2. Te Whakawehewehenga me te Whakawehewehenga Rōpū:
Ka whakamahia tēnei tikanga ina taea te wehewehe i te pūrau ki ētahi rōpū, kātahi ka tauwehe i ia rōpū. Hei tauira, ka taea te tauwehe i te pūrau \( x^3 – 6x^2 + 11x – 6 \) penei:
\[ x^3 – 6x^2 + 11x – 6 = (x-2)(x-3)(x-1) \]
3. Te Whakawehewehenga me te Ariā Toenga:
Ka whakamahia e tēnei tikanga te ariā toenga hei kimi i ngā pūtake o tētahi pūrinōmia, ā, ka whakamahia ēnei hei kimi i ngā tauwehe.
Kaihanga Pūtake (Kore Pūtake)
Ko te pūtake kore, te pūtake rānei o te pūrau he uara o \( x \) e ōrite ai te pūrau ki te kore. Arā, ko \( x \) he otinga ki te whārite pūrau \( P(x) = 0 \). Mena he pūrau tā tātou \( P(x) = a_n x^n + … + a_0 \), ko te kimi i te pūtake kore ko te tikanga kei te rapu tātou i te uara o \( x \) kia rite ai:
\[ a_n x^n + a_{n-1} x^{n-1} + … + a_1 x + a_0 = 0 \]
Te Kaupapa Taketake o te Āraipū
E ai ki te ariā taketake o te taurangi, kotahi te pūtake o ngā tau matatini o ia pūrau kore-pūmau. Ko te tikanga o tēnei, he n ngā pūtake o tētahi pūrau he tohu n mēnā ka tatauhia ngā pūtake e pā ana ki ō rātou tau maha.
Tikanga mō te Kimi i ngā Pūtake o tētahi Poronomial
1. Te Whakataurite:
Mena ka taea e tātou te tauwehe i tētahi pūrau, ka taea e tātou te kimi i ōna pūtake. Hei tauira, mā te whakamahi i te tauira i runga ake nei, mena kei a tātou \( P(x) = x^2 – 5x + 6 \), ka taea e tātou te tauwehe hei \( (x-2)(x-3) \). Mai i tēnei, ka mōhio tātou ko ngā pūtake ko \( x = 2 \) me \( x = 3 \).
2. Te Ariā Toenga me te Tikanga Wehewehenga Hanga:
He tikanga ā-ringa ake tēnei mō te kimi pūtake. E kī ana te ariā toenga mēnā ka wehea e tātou te pūrau \( P(x) \) ki \((xc)\), ko te toenga ko \( P(c) \). Mēnā ko \( P(c) = 0 \), ko \( (xc) \) he tauwehe o te pūrau, ā, ko \( c \) he pūtake o te pūrau.
3. Tikanga Tau:
Mō ngā pūrinomiara he teitei te nekehanga, mō ngā mea rānei e kore e taea te tauwehe ngāwari, ka whakamahia ngā tikanga tau pēnei i te tikanga Newton-Raphson hei whakatau tata i te otinga.
4. Tātai Tapawhā:
Mō te pūrau tapawhā \( ax^2 + bx + c = 0 \), ka kitea ngā pūtake mā te whakamahi i te tātai tapawhā:
\[ x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a} \]
5. Te Kaupapa Pūtake Whakaaroaro:
Mō ngā pūrinōmia me ngā tauwehenga whaitake, ka homai e tēnei ariā he rārangi o ngā pūtake whaitake pea ka taea te whakamātautau.
Te Hononga i waenga i ngā Tauwehe me ngā Pūtake o ngā Poronomia
He hononga tika kei waenganui i ngā tauwehe me ngā pūtake o tētahi pūrau. Mena ko \( r \) he pūtake o te pūrau \( P(x) \), ko \( (x – r) \) he tauwehe o \( P(x) \). I tua atu, ki te taea te tauwehe i \( P(x) \) hei \( (x – r)Q(x) \), ko \( r \) he pūtake o te pūrau.
Ko tētahi hua nui o tēnei whanaungatanga ko te taea te tauwehe i tētahi pūrau ki tētahi āhua rārangi ina tauwehea katoatia ki te papa matatini. Hei tauira, ko te pūrau pūtoru \( P(x) = x^3 – 6x^2 + 11x – 6 \) ka taea te tauwehe hei \( (x – 1)(x – 2)(x – 3) \), ko 1, 2, me te 3 ōna pūtake.
Ngā Tauira Taupānga
Tauira 1: Pūrau Tapawhā
Te kimi i ngā tauwehe me ngā pūtake o te pūrau \( P(x) = x^2 – 4x + 4 \):
1. Te Whakataurite:
Ka tautuhia e mātou a \( P(x) \) hei tapawhā tino tika:
P(x) = (x – 2)^2
2. Ngā Pūtake:
Mai i te whakawehewehe ka whiwhi tātou:
\( x – 2 = 0 \Pere Matau x = 2 \)
Nō reira, ko te pūtake o \( P(x) \) ko \( x = 2 \) me te maha o ngā tau 2.
Tauira 2: Pūrau Pūtoru
Te kimi i ngā tauwehe me ngā pūtake o te pūrau \( P(x) = x^3 – 6x^2 + 11x – 6 \):
1. Te Whakataurite:
Mā te whakamātau i ētahi uara mō x, ka kitea e tātou:
\[ P(1) = 1 – 6 + 11 – 6 = 0 \]
Nō reira, ko \( x = 1 \) he pūtake. Kātahi, ka taea e tātou te tuhi:
\[ P(x) = (x – 1)Q(x) \]
Ko Q(x) te wāhanga o te wehewehenga o \( P(x) \) ki \( (x – 1) \):
\[ Q(x) = x^2 – 5x + 6 \]
Kātahi ka haere tonu tātou ki te whakarea i te \( Q(x) \):
\[ Q(x) = (x – 2)(x – 3) \]
Nō reira,
\[ P(x) = (x – 1)(x – 2)(x – 3) \]
2. Ngā Pūtake:
Ko ngā pūtake o \( P(x) \) ko \( x = 1, 2, \) me \( 3 \).
Whakamutunga
He wāhanga nui ngā pūrau-ira o te pāngarau, ā, he maha ngā whakamahinga i roto i te pūtaiao me te hangarau. Ko te mārama ki ngā tauwehe me ngā kore o ngā pūrau-ira te mea nui ki te whakaoti rapanga maha e pā ana ki ngā pūrau-ira. He mea nui ngā tikanga tauwehe me ngā tikanga kimi pūtake mō te tātari pūrau-ira matatau. Mā te mārama pai, ka taea e tātou te whakahaere i ngā pūrau-ira kia pai ake te mahi me te tika.
