Te Whakamahi i te Ariā Toenga i roto i te Pāngarau
Ko te ariā toenga he ariā pāngarau e noho hei pou matua i roto i ngā momo peka pāngarau, tae atu ki te arapū, te ariā tau, me te pāngarau motumotu. Ehara i te mea he mea nui tēnei ariā i te taumata tuatahi anake, engari he nui hoki ōna tono i roto i te rangahau me te whanaketanga pāngarau matatau. Ka tūhuratia e tēnei tuhinga te ariā toenga me te hohonu, e kapi ana i tōna whakamāramatanga, ngā tono, me ētahi tauira hei mārama ki tana mahi i roto i ngā horopaki rerekē.
Te Mārama ki te Ariā Toenga
He ariā te ariā toenga i roto i te matawai pūrau. E kī ana tēnei ariā mēnā ka wehea he pūrau \( P(x) \) ki te pūrua \( (x – c) \), ko te toenga ko \( P(c) \). Ara, mō te pūrau \( P(x) \) mēnā ka wehea e tātou \( P(x) \) ki \( x – c \), ka whiwhi tātou i te āhua e whai ake nei:
\[ P(x) = (x – c)Q(x) + R \]
ko \( Q(x) \) te haurua pūronomial, ā, ko \( R \) te toenga. E ai ki te Remaine Theorem, ko \( R \) te uara o te mahi pūronomial ina \( x = c \), i roto rānei i te tuhi pāngarau:
\[ R = P(c) \]
Te Taunakitanga o te Ariā Toenga
Hei mārama ake i tēnei ariā, me whakamātau poto tātou. Mehemea he pūrau tā tātou \( P(x) \) ā, ka wehea e tātou mā \( (x – c) \). Kātahi ka taea e tātou te tuhi i tēnei:
\[ P(x) = (x – c)Q(x) + R \]
ko \( R \) te toenga o te wehenga. Nā te mea he binomial tohu-tuatahi a \( (x – c) \) , me noho pūmau te toenga \( R \) (nā te mea me iti iho te tohu o te toenga i te tohu o te wehewehe). Me whakakapi \( x = c \):
P(c) = (c – c)Q(c) + R
\[ P(c) = 0 \cdot Q(c) + R \]
\[ P(c) = R \]
Nō reira, kua whakamātauhia ko te toenga \( R \) he ōrite ki \( P(c) \).
Tauira o te Whakamahi i te Ariā Toenga
Me titiro tātou ki tētahi tauira tuturu o te ariā toenga kia mārama ai tātou ki tōna whakamahinga.
Tauira 1:
Me kī he pūrau tā tātou \( P(x) = x^3 – 4x^2 + 6x – 24 \). Me wehe tēnei pūrau mā \( x – 2 \).
Ko te taahiraa tuatahi ko te kimi i te uara o \( P(2) \):
\[ P(2) = 2^3 – 4 \cdot 2^2 + 6 \cdot 2 – 24 \]
\[ P(2) = 8 – 16 + 12 – 24 \]
\[ P(2) = -20 \]
Nō reira, ko te toenga o te wehewehe i te \( P(x) \) ki te \( x – 2 \) he -20.
Tauira 2:
Me kī he pūrau tā tātou \( P(x) = 2x^4 + 3x^3 – x + 5 \). Me wehe tēnei pūrau mā \( x + 1 \).
Ko te taahiraa tuatahi ko te kimi i te uara o \( P(-1) \):
\[ P(-1) = 2(-1)^4 + 3(-1)^3 – (-1) + 5 \]
P(-1) = 2(1) + 3(-1) + 1 + 5
\[ P(-1) = 2 – 3 + 1 + 5 \]
\[ P(-1) = 5 \]
Nō reira, ko te toenga o te wehewehe i te \( P(x) \) ki te \( x + 1 \) ko te 5.
Ngā Whakamahinga o te Ariā Toenga
He maha ngā whakamahinga o te ariā toenga i roto i ngā momo mara o te pāngarau. Ko ētahi o ngā whakamahinga matua ko:
1. Ngā Tauwehe Pūronomia:
Mena ko \( P(c) = 0 \), ko \( x – c \) he tauwehe o \( P(x) \). Ka āwhina tēnei ki te tauwehe i ngā pūrinomia nui ake, me ngā pūrinomia uaua ake.
2. Aromatawai Pūrau:
Mā te whakamahi i te ariā toenga, ka taea e tātou te aromatawai tere i te uara o te pūrau i tētahi pūwāhi kua hoatu, me te kore e hiahiatia te mahi i te wehewehenga roa.
3. Pūnaha Whakaiti:
I roto i te ariā tau me ngā rauropi, ka whakamahia te ariā toenga hei tiki tere i ngā toenga, he mea whai hua i roto i ngā tangohanga taupū me ngā tātaitanga e uru ana ngā tau nui.
4. Te Whakamātautau Pūtake:
E whakamahia ana tēnei ariā hei whakamātautau i ngā pūtake o ngā pūrinōmiara, koinei te pūtake o ētahi rauropi tau i roto i te rorohiko pūtaiao.
Te ariā toenga Hainamana
Haunga te ariā toenga i roto i te horopaki o ngā pūrinomia, kei reira anō te "Whakaaro Toenga Hainamana" e whānuitia ana te whakamahinga i roto i te ariā tau.
Me kī kei a tātou ētahi whārite ōrite:
\[ x \equiv a_1 \ (\text{mod} \n_1) \]
\[ x \equiv a_2 \ (\text{mod} \n_2) \]
\[ \vdots \]
\[ x \equiv a_k \ (\text{mod} \n_k) \]
Ina ko \( n_1, n_2, \ldots, n_k \) he takirua tau takirua takirua (he takirua tau kāore he tauwehe noa atu i te 1), ka whakamanahia e te Chinese Remain Theorem te noho o tētahi otinga ahurei modulo \( N \), ina ko \( N \) te hua o \( n_1, n_2, \ldots, n_k \).
Ngā Tauira o te Whakamahi i te Ariā Toenga Hainamana
Me kī kei a tātou te pūnaha ōrite e whai ake nei:
\[ x \equiv 2 \ (\text{mod} \ 3) \]
\[ x \equiv 3 \ (\text{mod} \ 5) \]
\[ x \equiv 2 \ (\text{mod} \ 7) \]
Me kimi e tātou he uara o x e tutuki ai ēnei whārite katoa. Nā te mea he coprime te 3, te 5, me te 7, ka taea e tātou te whakamahi i te Chinese Remain Theorem.
Ko te taahiraa tuatahi ko te tatau i te \( N \):
\[ N = 3 \whakanuia te 5 \whakanuia te 7 = 105 \]
Ko te taahiraa tuarua ko te tatau i te \( N_i \) mō ia moduli:
\[ N_1 = \frac{N}{3} = 35 \]
\[ N_2 = \frac{N}{5} = 21 \]
\[ N_3 = \frac{N}{7} = 15 \]
Ko te taahiraa tuatoru ko te kimi i te whakahuri whakarea o \( N_i \) modulo ngā moduli e rite ana:
\[ 35x \equiv 1 \ (\text{mod} \ 3) \e kī ana ko x = 2 \]
\[ 21x \equiv 1 \ (\text{mod} \ 5) \e kī ana ko x = 1 \]
\[ 15x \equiv 1 \ (\text{mod} \ 7) \e kī ana ko x = 1 \]
Na ka whakakotahi i ngā mea katoa:
\[ x = a_1N_1x_1 + a_2N_2x_2 + a_3N_3x_3 \]
\[ x = 2 \cdot 35 \cdot 2 + 3 \cdot 21 \cdot 1 + 2 \cdot 15 \cdot 1 \]
\[ x = 140 + 63 + 30 = 233 \]
Hei whakamutunga, ka tangohia e mātou te modulo N:
\[ x \equiv 233 \ (\text{mod} \ 105) \]
\[ x = 233 – 2 \cdot 105 \]
\[ x = 23 \]
Nō reira, ko te otinga o te pūnaha ōritetanga ko \( x = 23 \).
Whakamutunga
He taputapu kaha, he taputapu maha hoki te ariā toenga i roto i te pāngarau me te ariā tau. Mēnā he mārama pai, ka taea e ia te tere ake i ngā tataunga uaua, me te whakatakoto huarahi mō ētahi atu tātaritanga i roto i te pāngarau. Kei roto i ōna whakamahinga ko te aromatawai pūrau, te whakawehewehe, ngā rauropi tauoti, me te whakaoti rapanga ōrite, e kitea ana i roto i te ariā toenga Hainamana. Mā te ako i tēnei ariā, ka taea e tātou te whakapai ake i tō tātou kaha ki te whakaoti rapanga pāngarau maha me te whai hua ake.
