Tusaalooyinka Su'aalaha iyo Doodaha ku saabsan Polynomials iyo Polynomial Functions
Pendahuluan
Bolinimo iyo shaqooyin bolinimo waa mowduucyo muhiim ah oo ku jira xisaabta kuwaas oo si joogto ah uga soo muuqda codsiyada sayniska iyo injineernimada kala duwan. Bolinimo waa tibaax xisaabeed oo ka kooban doorsoomayaal, joogto, iyo hawlgallada isku darka, kala-goynta, iyo isku dhufashada, waxayna leedahay jilayaal aan taban ahayn. Tusaale fudud oo bolinimo waa \( P(x) = x^2 + 2x + 1 \). Shaqada bolinimo waa shaqo lagu muujiyay qaab bolinimo. Maqaalkan, waxaan ka wada hadli doonnaa tusaalooyin bolinimo iyo shaqooyin bolinimo, iyo sharraxaaddooda faahfaahsan.
Qeexitaanka Polynomial
Bolinomial hal doorsoome ah \( x \) waxaa lagu qeexi karaa qaabka guud sida soo socota:
\[ P(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \]
Halkee:
– \( a_n, a_{n-1}, \ldots, a_1, a_0 \) waa isku-dhafnaanno ah tirooyin dhab ah.
– \( n \) waa awoodda ugu sarreysa ee ah tiro aan taban ahayn.
Su'aalo iyo Doodo Tusaale ah
Su'aal Tusaale ah 1: Xisaabinta Qiimaha Polynomial
Su'aal:
Marka la eego polynomial \( P(x) = 3x^3 – 2x^2 + 4x – 5 \). Xisaabi qiimaha \( P(2) \).
Dood:
Si loo xisaabiyo qiimaha polynomial-ka ee \( x = 2 \), waxaan ku beddeleynaa \( x \) 2 polynomial-ka:
\[ P(2) = 3(2)^3 – 2(2)^2 + 4(2) – 5 \]
\[ P(2) = 3 \cdot 8 - 2 \cdot 4 + 4 \cdot 2 - 5 \]
\[ P(2) = 24 – 8 + 8 – 5 \]
\[ P(2) = 19 \]
Markaa, qiimaha \( P(2) \) waa 19.
Su'aal Tusaale 2aad: Helitaanka Xididdada Polynomial
Su'aal:
Soo hel xididdada polynomial-ka \( P(x) = x^2 – 5x + 6 \).
Dood:
Waxaan isticmaalnaa habka kala-soocidda si aan u helno xididdada polynomial-ka:
\[ P(x) = x^2 – 5x + 6 \]
\[ P(x) = (x – 2)(x – 3) \]
Markaa, xididdadu waa \( x = 2 \) iyo \( x = 3 \).
Su'aal Tusaale ah 3: Xisaabinta Noocyada Kala-soocidda Polynomial
Su'aal:
Marka la eego polynomial-ka \( P(x) = 4x^3 – 3x^2 + 2x – 1 \). Xisaabi waxyaabaha ka soo jeeda polynomial-ka koowaad iyo labaad.
Dood:
Qodobkii ugu horreeyay ee ka soo jeeda polynomial-ka \( P(x) \) waa:
\[ P'(x) = \frac{d}{dx}(4x^3 – 3x^2 + 2x – 1) \]
\[ P'(x) = 12x^2 – 6x + 2 \]
Qodobbada labaad ee ka soo jeeda polynomial-ka \( P(x) \) waa:
\[ P”(x) = \frac{d}{dx}(12x^2 – 6x + 2) \]
\[ P”(x) = 24x – 6 \]
Markaa, beddelka koowaad ee \( P(x) \) waa \( 12x^2 – 6x + 2 \) halka beddelka labaadna uu yahay \( 24x – 6 \).
Su'aal Tusaale ah 4: Helitaanka Shaqada Polynomial-ka ee laga helo Dhibcaha La Bixiyay
Su'aal:
Soo hel shaqada polynomial-ka heerka labaad \( P(x) \) ee dhex marta dhibcaha (1, 2), (2, 3), iyo (3, 14).
Dood:
Waxaan u qaadaneynaa shaqada polynomial-ka heerka labaad ee foomka:
\[ P(x) = ax^2 + bx + c \]
Iyadoo dhibcaha lagu beddelayo polynomial-ka:
1) Laga bilaabo (1, 2): \( a(1)^2 + b(1) + c = 2 \) \(\rightarrow a + b + c = 2 \)
2) Laga bilaabo (2, 3): \( a(2)^2 + b(2) + c = 3 \) \(\rightarrow 4a + 2b + c = 3 \)
3) Laga bilaabo (3, 14): \( a(3)^2 + b(3) + c = 14 \) \(\rightarrow 9a + 3b + c = 14 \)
Marka xigta waxaan haynaa nidaam isle'egyo toosan:
\[ a + b + c = 2 \]
\[ 4a + 2b + c = 3 \]
\[ 9a + 3b + c = 14 \]
Waxaan xallineynaa nidaamkan isle'egyada:
1) Ka jar isle'egyada labaad iyo kan koowaad:
\[ (4a + 2b + c) – (a + b + c) = 3 – 2 \]
\[ 3a + b = 1 \]
2) Ka jar isle'egyada saddexaad iyo labaad:
\[ (9a + 3b + c) - (4a + 2b + c) = 14 - 3 \]
\[ 5a + b = 11 \]
Waxaan xallinaa nidaamka isle'egyada:
\[ 3a + b = 1 \]
\[ 5a + b = 11 \]
Ka jar isle'egyada labaad iyo kan koowaad:
\[ (5a + b) - (3a + b) = 11 - 1 \]
\[ 2a = 10 \]
\[ a = 5 \]
Ku beddel \( a = 5 \) mid ka mid ah isle'egyada:
\[ 3(5) + b = 1 \]
\[ 15 + b = 1 \]
\[ b = -14 \]
Ku beddel \( a = 5 \) iyo \( b = -14 \) mid ka mid ah isle'egyada asalka ah:
\[ 5 + (-14) + c = 2 \]
\[ -9 + c = 2 \]
\[ c = 11 \]
Markaa, shaqada polynomial-ka ee dhex marta qodobbadan waa:
\[ P(x) = 5x^2 – 14x + 11 \]
Xiritaanka
Maqaalkan, waxaan ka wada hadalnay dhowr tusaale oo dhibaatooyin ah oo ku lug leh polynomials iyo polynomials, iyo sidoo kale sida loo xalliyo. Dhibaatooyinkani waxay u dhexeeyaan xisaabinta qiimaha polynomial, helitaanka xididdada polynomial, xisaabinta soo-saarka polynomial, ilaa helitaanka shaqo polynomial ah oo laga soo bilaabo dhibco la yaqaan. Polynomials iyo hawlaha polynomial waa aasaaska fikrado badan oo xisaabeed oo horumarsan, sida falanqaynta tirooyinka, aljabrada toosan, iyo aragtida tirada. Fahmidda aasaaskan ayaa muhiim u ah guusha dhinacyo kala duwan oo tacliimeed iyo xirfadeed ah.
