Ngā Tauira Pātai e Matapaki ana i ngā Tuakiri Pūrau
He ariā taketake ngā tuakiri pūrau i roto i te pāngarau, e whakamahia ana hei whakangawari i ngā kīanga pāngarau me te whakaoti rapanga. I roto i tēnei tuhinga, ka matapakihia e mātou ētahi tauira rapanga me ngā otinga e pā ana ki ngā tuakiri pūrau hei whakahōhonu ake i tō mātou māramatanga ki te kaupapa. Ka tīmata mātou me te whakamāramatanga, kātahi ka neke atu ki ngā tauira rapanga me ō rātou otinga.
Te Whakamāramatanga o te Tuakiri Poronomi
Ko te tuakiri pūrau he whārite e mau ana mō ngā uara katoa o ngā taurangi. Hei tauira, ko te tuakiri pūrau rongonui ko:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
E mau ana tēnei tuakiri mō ngā uara katoa o \( a \) me \( b \). He maha atu anō ngā tuakiri nui i roto i te arapū, pērā i:
\[ (a – b)^2 = a^2 – 2ab + b^2 \]
\[ a^2 – b^2 = (a – b)(a + b) \]
Nā, me titiro tātou ki ētahi tauira rapanga hei whakamārama i te whakamahinga o ngā tuakiri pūrau.
Ngā Pātai Tauira me te Kōrero
Tauira 1: Te Whakangāwari i tētahi Kīanga
Pātai:
Whakangāwaritia ngā kīanga e whai ake nei mā te whakamahi i ngā tuakiri pūrau:
\[ (2x + 3y)^2 \]
Kōrero:
Ka whakamahia e mātou te tuakiri pūrau taketake:
\[ (a + b)^2 = a^2 + 2ab + b^2 \]
Anei, \( a = 2x \) me \( b = 3y \). Mā te whakakapi i ēnei uara ki te tuakiri ka whiwhi tātou:
\[ (2x + 3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2 \]
\[ = 4x^2 + 12xy + 9y^2 \]
Nā, ko te kīanga māmā ko:
\[ 4x^2 + 12xy + 9y^2 \]
Tauira 2: Whārite Tuakiri
Pātai:
Whakamātauria ngā tuakiri pūrau e whai ake nei:
\[ (x – y)^2 + (x + y)^2 = 2(x^2 + y^2) \]
Kōrero:
Ka whakawhanuitia e tātou ngā taha e rua o te whārite, kia kite ai mēnā he ōrite ngā kīanga e rua.
Tirohia te taha maui:
\[ (x – y)^2 + (x + y)^2 \]
Whakamahia ngā tuakiri \( (a – b)^2 \) me \( (a + b)^2 \):
\[ = (x^2 – 2xy + y^2) + (x^2 + 2xy + y^2) \]
Whakakotahitia ngā kīanga e rua:
\[ = x^2 – 2xy + y^2 + x^2 + 2xy + y^2 \]
\[ = x^2 + x^2 + y^2 + y^2 \]
\[ = 2x^2 + 2y^2 \]
Kua whakamāmāhia te taha maui ki \( 2(x^2 + y^2) \), he ōrite ki te taha matau. Nō reira, kua whakamātauhia tēnei tuakiri.
Tauira 3: Te Whakawehewehenga o ngā Poronomia
Pātai:
Tātaitia ngā pūrinomia e whai ake nei:
\[ x^4 – 16 \]
Kōrero:
Ka taea e tātou te whakamahi i te tuakiri \( a^2 – b^2 = (a – b)(a + b) \). I konei, kia mōhio koe ka taea te tuhi i te \( x^4 \) hei \( (x^2)^2 \):
\[ x^4 – 16 = (x^2)^2 – 4^2 \]
Whakamahia te tuakiri:
\[ = (x^2 – 4)(x^2 + 4) \]
Heoi anō, ka taea tonu te tāpiri i te \( x^2 – 4 \) nā te mea:
\[ x^2 – 4 = (x – 2)(x + 2) \]
Nō reira, ko te whakawehewehenga katoa ko:
\[ x^4 – 16 = (x – 2)(x + 2)(x^2 + 4) \]
Tauira 4: Ngā Poronomia Teitei Ake
Pātai:
I runga i ngā tuakiri pūrau e whai ake nei:
\[ x^5 – 1 = (x – 1)(x^4 + x^3 + x^2 + x + 1) \]
Whakamātauria te tuakiri.
Kōrero:
Ka whakamāramahia tēnei mā te mahi wehewehe pūrau. Ko tēnei tikanga ko te wehewehe i te \( x^5 – 1 \) mā te \( x – 1 \) kātahi ka manatoko he kore tūturu te toenga.
Whakamahi i te wehenga pūrau:
1. Wehea ngā kupu teitei rawa \( x^5 \) ki \( x \) kia whiwhi ai i te kupu tuatahi \( x^4 \).
2. Whakareatia te \( x^4 \) ki te \( x – 1 \) ka tangohia te hua mai i te \( x^5 – 1 \).
3. Whakahokia tēnei tukanga kia tangohia rā anō ngā kupu katoa.
I muri i te mahi i te wehenga, ka whiwhi tātou:
\[ x^5 – 1 \div (x-1) = x^4 + x^3 + x^2 + x + 1 \]
Nā te mea kāore he toenga, e whakaatu ana tēnei:
\[ x^5 – 1 = (x – 1)(x^4 + x^3 + x^2 + x + 1) \]
Tauira 5: Ngā Pūrinomia me ngā Pūtake Matatini
Pātai:
Mena he tauwehe o tētahi pūrau-ā-ira a \( x + 1 \) \( f(x) \), kimihia ngā pūtake kē atu o te pūrau-ā-ira kua hoatu \( f(x) = x^3 + x^2 – 6x – 6 \).
Kōrero:
Ina he tauwehe a \( x + 1 \) o \( f(x) \), ko te tikanga ko \( x = -1 \) tētahi o ngā pūtake o te pūrau.
Whakahaerehia te Wehenga Pūrau Tika:
1. Wehea te \( f(x) \) ki te \( x + 1 \) mā te whakamahi i te tikanga wehewehe roa, te tikanga wehewehe hangai rānei.
2. Whakaitihia te pūrinōmia mā te tau i whiwhihia.
I muri i te mahi wehewehenga hangai, ka whiwhi tātou:
\[ f(x) = (x + 1)(x^2 – 6) \]
Ka taea te wehewehe anō i a \( x^2 – 6 \) ki:
\[ x^2 – 6 = (x – \sqrt{6})(x + \sqrt{6}) \]
Nō reira, ko ngā pūtake o te pūrau ko:
\[ x = -1, \; x = \sqrt{6}, \; x = -\sqrt{6} \]
Mā ngā tauira i runga ake nei, kua mārama tātou ki te whakamahinga o ngā tuakiri pūrau i roto i te whakangawari i ngā kīanga, te whakamatau i ngā whārite, te tauwehe i ngā pūrau, me te kimi i ngā pūtake o ngā pūrau.
Whakamutunga
He mea nui te mahi a ngā tuakiri pūrau i roto i te pāngarau, te whakangawari i ngā kīanga pāngarau, te tauwehe i ngā pūrau, me te whakaoti whārite. Mā te mārama me te whakamahi i ngā tuakiri pūrau ka āwhina i a tātou ki te whakaoti rapanga pāngarau me te whai hua. Ko te tumanako, mā ngā tauira kua kōrerohia i roto i tēnei tuhinga ka kitea he māramatanga hohonu ake mō ngā tuakiri pūrau me ō rātou whakamahinga.
