Ngā Tauira Pātai e Matapaki ana i te Tāpiri, te Tango, me te Whakarea o ngā Poronomia
He wāhanga nui ngā pūrau-ira o te pāngarau me te pāngarau whānui. Ko ngā pūrau-ira he kupu kotahi, neke atu rānei, ko ia kupu he pūmau, he taurangi rānei kua whakanuia ki te mana. Ka taea te whakakotahi i ngā pūrau-ira mā te whakamahi i ngā mahi taketake pēnei i te tāpiri, te tango, me te whakarea. Ka matapakihia e tēnei tuhinga ngā tauira rapanga me pēhea te whakaoti rapanga tāpiri, tango, me te whakarea o ngā pūrau-ira.
Te Tāpiri i ngā Poronomia
Ko te tāpiri i ngā pūrinōmia ko te tāpiri i ngā taunga o ngā kupu ōrite. Anei ngā mahi me ngā tauira rapanga hei āwhina i a koe ki te mārama ki te tāpiri i ngā pūrinōmia.
Tauira Pātai 1:
Tāpirihia ngā pūrinōmiora e whai ake nei: \( (3x^2 + 2x + 5) \) me \( (4x^2 – x + 7) \).
Ngā Hipanga Whakaoti:
1. Tuhia ngā pūrinomia e rua hei tāpiri:
\[
(3x^2 + 2x + 5) + (4x^2 – x + 7)
\]
2. Whakarōpūhia ngā iwi ōrite:
\[
(3x^2 + 4x^2) + (2x – x) + (5 + 7)
\]
3. Tāpirihia ngā tauwehenga o ngā kupu ōrite:
\[
7x^2 + x + 12
\]
Nō reira, ko te hua o te tāpiri i ngā pūrinōmia ko \( 7x^2 + x + 12 \).
Tangohanga Pūrau
He rite tonu te tikanga o te tango i ngā pūrinōmia ki te tāpiri, engari ka tangohia ngā tauwehenga o ngā kupu ōrite. Anei tētahi tauira raruraru me ngā mahi hei whakaoti.
Tauira Pātai 2:
Tangohia te pūrau e whai ake nei: \( (5x^3 + 3x^2 + 4x) \) mā \( (2x^3 + x^2 – 3x) \).
Ngā Hipanga Whakaoti:
1. Tuhia ngā pūrinoma e rua hei tango:
\[
(5x^3 + 3x^2 + 4x) – (2x^3 + x^2 – 3x)
\]
2. Whakarōpūhia ngā iwi ōrite:
\[
(5x^3 – 2x^3) + (3x^2 – x^2) + (4x – (-3x))
\]
3. Tangohia ngā tauwehenga mai i ngā kupu ōrite:
\[
3x^3 + 2x^2 + 7x
\]
Nō reira, ko te hua o te tango i ngā pūrinōmia ko \( 3x^3 + 2x^2 + 7x \).
Whakareatanga Pūrau
He uaua ake te whakarea i ngā pūrau-ira nā te mea me tohatoha ia kupu i roto i tētahi pūrau-ira ki ia kupu i roto i tētahi atu. Anei ngā mahi me ngā tauira rapanga hei āwhina i a koe ki te mārama ki te whakarea pūrau-ira.
Tauira Pātai 3:
Whakareatia ngā pūrau e whai ake nei: \( (2x + 3) \) me \( (x^2 – x + 4) \).
Ngā Hipanga Whakaoti:
1. Tuhia ngā pūrau e rua hei whakarea:
\[
(2x + 3)(x^2 – x + 4)
\]
2. Tohaina ia wāhanga o te pūrau tuatahi ki ia wāhanga o te pūrau tuarua:
\[
2x(x^2 – x + 4) + 3(x^2 – x + 4)
\]
3. Whakareatia ia wāhanga:
\[
2x \cdot x^2 = 2x^3
\]
\[
2x \cdot (-x) = -2x^2
\]
\[
2x \cdot 4 = 8x
\]
\[
3 \cdot x^2 = 3x^2
\]
\[
3 \cdot (-x) = -3x
\]
\[
3 \cdot 4 = 12
\]
4. Kohia ngā hua katoa:
\[
2x^3 – 2x^2 + 8x + 3x^2 – 3x + 12
\]
5. Whakakotahitia, whakarōpūtia hoki ngā kupu ōrite:
\[
2x^3 + (-2x^2 + 3x^2) + (8x – 3x) + 12
\]
6. Whakangāwaritia:
\[
2x^3 + x^2 + 5x + 12
\]
Nō reira, ko te hua o te whakarea i ngā pūrau ko \( 2x^3 + x^2 + 5x + 12 \).
Ngā Tauira Pātai Tāpiri:
Tauira Pātai 4:
Whakareatia ngā pūrau e whai ake nei: \( (x + 2) \) me \( (x^2 + 2x + 1) \).
Ngā Hipanga Whakaoti:
1. Tuhia ngā pūrau e rua hei whakarea:
\[
(x + 2)(x^2 + 2x + 1)
\]
2. Tohaina ia wāhanga o te pūrau tuatahi ki ia wāhanga o te pūrau tuarua:
\[
x(x^2 + 2x + 1) + 2(x^2 + 2x + 1)
\]
3. Whakareatia ia wāhanga:
\[
x \cdot x^2 = x^3
\]
\[
x \cdot 2x = 2x^2
\]
\[
x \cdot 1 = x
\]
\[
2 \cdot x^2 = 2x^2
\]
\[
2 \cdot 2x = 4x
\]
\[
2 \cdot 1 = 2
\]
4. Kohia ngā hua katoa:
\[
x^3 + 2x^2 + x + 2x^2 + 4x + 2
\]
5. Whakakotahitia, whakarōpūtia hoki ngā kupu ōrite:
\[
x^3 + (2x^2 + 2x^2) + (x + 4x) + 2
\]
6. Whakangāwaritia:
\[
x^3 + 4x^2 + 5x + 2
\]
Nō reira, ko te hua o te whakarea i ngā pūrau ko \( x^3 + 4x^2 + 5x + 2 \).
Ngā Māramatanga Tāpiri
1. Te Whakamahi i ngā Tuakiri Pūrau: I ngā wā maha, mā te mārama ki ngā tuakiri taketake pēnei i te \( (a+b)^2 = a^2 + 2ab + b^2 \) me te \( (ab)^2 = a^2 – 2ab + b^2 \) ka tere ake te tātaitanga.
2. Ngā Hapa Noa: I te tāpiri, i te tango rānei i ngā pūrinōmia, me whakarōpū tonu i ngā kupu o te nekehanga kotahi. Ko ngā hapa whakarōpū te tino take o ngā hua hē.
3. Te Whakareatanga Whakarōpū (Tohatoha): I te wā e mahi ana me te whakareatanga pūronomial, kia maumahara tonu ki te tohatoha tika i ia kupu puta noa i ngā taurangi katoa. Mā te kore aro ki tētahi kupu ka pakaru te whakautu katoa.
Whakamutunga
He mea nui ngā pūrau-ira i roto i te pāngarau, ā, he mea nui te mārama ki ēnei mō ngā ākonga me ngā tohunga e mahi ana i roto i te hangarau, te ahupūngao, me ētahi atu pūtaiao. Mā te mārama me te whakaharatau pinepine i te tāpiri, te tango, me te whakarea i ngā pūrau-ira, ka taea e te tangata te mahi tere i ngā tataunga uaua ake i roto i ngā horopaki pāngarau maha. Ko te tumanako ka āwhina ngā tauira kua whakaratohia i ngā kaipānui ki te mārama ake ki tēnei ariā taketake, me te whiwhi māia ki te whakaoti rapanga e pā ana ki ngā pūrau-ira.
