Ngā Tauira Pātai mō te Matapaki i te Wehenga Pūrau
He kaupapa nui te wehewehenga pūrau i roto i te pāngarau, inā koa te arapūrei. He maha ngā whakamahinga o ngā pūrau i roto i ngā momo mara pūtaiao, pērā i te ahupūngao, te ōhanga, me te hangarau, hei whakatauira i ngā āhuatanga uaua. Mā te wehewehe i ngā pūrau, ka taea e tātou te whakahaere i ngā raruraru kia māmā ake ai te mārama. Ka matapakihia e tēnei tuhinga te tikanga wehewehe i ngā pūrau, me ngā tauira raruraru me ngā kōrero.
1. Tikanga Wehenga Roa
Ko te tikanga tuatahi ka matapakihia e tātou ko te whakawehe roa, he rite ki te whakawehe roa mō ngā tau. He tikanga pūnaha, he tikanga taipitopito hoki, ā, he tino āwhina ki te mārama ki ngā kaupapa matua o te whakawehe pūrau.
Tauira raruraru:
Wehea \( 2x^3 + 3x^2 – 5x + 7 \) ki \( x + 1 \).
Ngā Hipanga:
1. Tuhia te pūrau hei wehewehe (wehenga) me te pūrau whakawehe (wehenga).
Te wehewehenga: \( 2x^3 + 3x^2 – 5x + 7 \)
Wehewehe: \( x + 1 \)
2. Wehea te wāhanga tuatahi o te wehewehenga ki te wāhanga tuatahi o te whakawehe.
Wehea te \( 2x^3 \) ki te \( x \) kia puta ai te \( 2x^2 \).
3. Whakareatia te whakawehenga ki te haurua.
\( (x + 1) \whakanuia 2x^2 = 2x^3 + 2x^2 \)
4. Tangohia te hua whakarea mai i te wehewehenga.
\( (2x^3 + 3x^2 – 5x + 7) – (2x^3 + 2x^2) = x^2 – 5x + 7 \)
5. Whakahokia ngā taahiraa 2 ki te 4 me te hua i tangohia hei wehewehenga hou.
– \( x^2 ÷ x = x \)
– \( (x + 1) \whakanuia x = x^2 + x \)
– \( (x^2 – 5x + 7) – (x^2 + x) = -6x + 7 \)
6. Me haere tonu te tukanga:
– \( -6x ÷ x = -6 \)
– \( (x + 1) \whakanuia -6 = -6x – 6 \)
– \( (-6x + 7) – (-6x – 6) = 13 \)
Ko te hua whakamutunga ko:
\[ 2x^2 + x – 6, \text{ me te toenga } 13 \]
Nō reira, \( \frac{2x^3 + 3x^2 – 5x + 7}{x + 1} = 2x^2 + x – 6 + \frac{13}{x+1} \).
2. Tikanga Wehewehenga Hangahanga
Ko te tikanga tuarua ko te wehewehe hangai, he tere ake, he pai ake hoki i te wehewehe roa, engari e pā ana ki te wehewehe mā te whakamahi i ngā pūrau o te āhua \( x – k \).
Tauira raruraru:
Wehea te \( 2x^3 + 3x^2 – 5x + 7 \) ki te \( x – 1 \).
Ngā Hipanga:
1. Whakakapia te whakahuri o te tauwehenga whakawehe.
Nā te mea ko te wehewehenga ko \( x – 1 \), ko te whakahurihanga ko \( 1 \).
2. Mātakihia ngā taunga o ngā pūrinomia hei wehewehe.
\( [2, 3, -5, 7] \)
3. Whakaotia te whakahiato:
– Whakaitihia te tauwehenga tuatahi: \( 2 \)
– Whakareatia te whakamuri o te whakawehenga \( 1 \) ki te uara hou, ka tāpiri atu ki te tauwehenga e whai ake nei.
– \[ 2 \]
– \( 2 \whakareatia ki te 1 = 2 \)
– \( 3 + 2 = 5 \)
– \[ 2, 5 \]
– \( 5 \whakareatia ki te 1 = 5 \)
– \(-5 + 5 = 0 \)
– \[ 2, 5, 0 \]
– \( 0 \whakareatia ki te 1 = 0 \)
– \( 7 + 0 = 7 \)
– \[ 2, 5, 0, 7 \]
Ko te hua whakamutunga ko:
\[ 2x^2 + 5x + 0, \text{ me te toenga } 7 \]
Nō reira, \( \frac{2x^3 + 3x^2 – 5x + 7}{x – 1} = 2x^2 + 5x + \frac{7}{x-1} \).
3. Te Wehewehenga mā ngā Poronomia Teitei Ake
E pā ana hoki te wehenga pūrau ki ngā wehewehe uaua ake.
Tauira raruraru:
Wehea \( x^4 – 3x^3 + 2x^2 – x + 5 \) ki \( x^2 – x + 1 \).
Ngā Hipanga:
1. Tuhia te wehewehenga me te wehewehenga.
Te wehewehenga: \( x^4 – 3x^3 + 2x^2 – x + 5 \)
Wehewehe: \( x^2 – x + 1 \)
2. Wehea te wāhanga tuatahi o te wehewehenga ki te wāhanga tuatahi o te whakawehe.
\( x^4 ÷ x^2 = x^2 \)
3. Whakareatia te whakawehenga ki te haurua.
\( (x^2 – x + 1) \whakanuia x^2 = x^4 – x^3 + x^2 \)
4. Tangohia te hua mai i te wehewehenga.
\( (x^4 – 3x^3 + 2x^2 – x + 5) – (x^4 – x^3 + x^2) = -2x^3 + x^2 – x + 5 \)
5. Whakahokia ngā taahiraa 2 ki te 4.
– \( -2x^3 ÷ x^2 = -2x \)
– \( (x^2 – x + 1) \whakanuia -2x = -2x^3 + 2x^2 – 2x \)
– \( (-2x^3 + x^2 – x + 5) – (-2x^3 + 2x^2 – 2x) = -x^2 + x + 5 \)
6. Me haere tonu te tukanga:
– \( -x^2 ÷ x^2 = -1 \)
– \( (x^2 – x + 1) \whakanuia -1 = -x^2 + x – 1 \)
– \( (-x^2 + x + 5) – (-x^2 + x – 1) = 6 \)
Ko te hua whakamutunga ko:
\[ x^2 – 2x – 1, \text{ me te toenga } 6 \]
Nō reira, \( \frac{x^4 – 3x^3 + 2x^2 – x + 5}{x^2 – x + 1} = x^2 – 2x – 1 + \frac{6}{x^2 – x + 1} \).
Whakamutunga
He pūkenga nui te wehewehe i ngā pūrau-ira hei ako mā ngā ākonga e ako ana i te pāngarau. E rua ngā tikanga matua—te wehewehe roa me te wehewehe hangai—e tuku ana i ngā huarahi rerekē, he rerekē hoki ōna painga me ōna ngoikoretanga. Ahakoa he pai te tikanga wehewehe roa mō ngā wehewehe uaua ake, ka whakaratohia e te tikanga wehewehe hangai he huarahi tere ake, he huarahi whai hua ake hoki ki te wehewehe mā ngā pūrau-ira o te āhua \( x – k \). Mēnā he nui te mahi, ka taea te whakamahi i te mārama ki ēnei ariā me ngā tikanga ki ngā momo raruraru pāngarau matatau ake.
