Siv Cov Theorem seem hauv Kev Ua lej
Tus theorem seem yog ib lub tswv yim lej uas feem ntau yog lub hauv paus tseem ceeb hauv ntau ceg ntawm lej, suav nrog algebra, lej theory, thiab lej discrete. Lub tswv yim no tsis yog tsuas yog cuam tshuam rau theem pib xwb tab sis kuj muaj cov ntawv thov tseem ceeb hauv kev tshawb fawb thiab kev txhim kho lej siab heev. Tsab xov xwm no yuav tshawb nrhiav tus theorem seem tob tob, npog nws cov lus txhais, kev siv, thiab ntau qhov piv txwv kom nkag siab tias nws ua haujlwm li cas hauv ntau qhov xwm txheej.
Nkag Siab Txog Qhov Theorem Uas Tseem Muaj Seem
Cov seem theorem yog ib qho theorem hauv polynomial algebra. Cov theorem no hais tias yog tias ib qho polynomial \(P(x) \) raug faib los ntawm binomial \((x - c) \), ces cov seem yog \(P(c) \). Ntawd yog, rau polynomial \(P(x) \) yog tias peb faib \(P(x) \) los ntawm \(x - c \), peb yuav tau txais daim ntawv hauv qab no:
\[ P(x) = (x – c)Q(x) + R \]
qhov twg \(Q(x) \) yog tus lej polynomial thiab \(R \) yog tus lej seem. Raws li Remainder Theorem, \(R \) yog tus nqi ntawm polynomial function thaum \(x = c \), lossis hauv lej cim:
\[ R = P(c) \]
Pov thawj ntawm qhov seem Theorem
Yuav kom nkag siab zoo dua txog qhov kev xav no, cia peb ua pov thawj nws luv luv. Xav tias peb muaj ib qho polynomial \( P(x) \) thiab peb faib nws los ntawm \( (x - c) \). Tom qab ntawd peb tuaj yeem sau tias:
\[ P(x) = (x – c)Q(x) + R \]
qhov twg \( R \) yog qhov seem ntawm qhov faib. Txij li thaum \( (x - c) \) yog thawj qib binomial, qhov seem \( R \) yuav tsum yog qhov tsis hloov pauv (vim tias qib ntawm qhov seem yuav tsum tsawg dua qib ntawm tus divisor). Cia peb hloov \( x = c \):
\[ P(c) = (c – c)Q(c) + R \]
\[ P(c) = 0 \cdot Q(c) + R \]
\[ P(c) = R \]
Yog li, nws tau ua pov thawj tias cov seem \(R\) yog sib npaug rau \(P(c)\).
Piv txwv ntawm Kev Siv Cov Theorem seem
Cia peb saib ib qho piv txwv ntawm cov theorem seem kom nkag siab nws daim ntawv thov.
Piv txwv 1:
Xav tias peb muaj ib tug polynomial \( P(x) = x^3 – 4x^2 + 6x – 24 \). Peb xav faib cov polynomial no los ntawm \( x – 2 \).
Kauj ruam thawj zaug yog nrhiav tus nqi ntawm \( P(2) \):
\[ P(2) = 2^3 – 4 \cdot 2^2 + 6 \cdot 2 – 24 \]
\[ P(2) = 8 – 16 + 12 – 24 \]
\[ P(2) = -20 \]
Yog li, qhov seem ntawm kev faib \(P(x)\) los ntawm \(x – 2\) yog -20.
Piv txwv 2:
Xav tias peb muaj ib tug polynomial \( P(x) = 2x^4 + 3x^3 – x + 5 \). Peb xav faib cov polynomial no los ntawm \( x + 1 \).
Kauj ruam thawj zaug yog nrhiav tus nqi ntawm \( P(-1) \):
\[ P(-1) = 2(-1)^4 + 3(-1)^3 – (-1) + 5 \]
\[ P(-1) = 2(1) + 3(-1) + 1 + 5 \]
\[ P(-1) = 2 – 3 + 1 + 5 \]
\[ P(-1) = 5 \]
Yog li ntawd, qhov seem ntawm kev faib \(P(x)\) los ntawm \(x + 1\) yog 5.
Cov Kev Siv ntawm Cov Theorem seem
Cov theorem seem muaj ntau yam kev siv rau ntau yam kev kawm lej. Qee qhov kev siv tseem ceeb suav nrog:
1. Cov Yam Ntxim Saib Polynomial:
Yog tias \( P(c) = 0 \), ces \( x – c \) yog ib qho ntawm \( P(x) \). Qhov no pab ua kom cov polynomials loj dua thiab nyuaj dua.
2. Kev Ntsuam Xyuas Polynomial:
Siv cov theorem seem, peb tuaj yeem soj ntsuam qhov nqi ntawm polynomial ntawm ib qho chaw uas tau muab yam tsis tas yuav ua kev faib ntev.
3. Kev Txo Algorithm:
Hauv kev xav txog tus lej thiab cov algorithms, cov theorem seem yog siv los kom tau txais cov seem sai sai, uas yog qhov muaj txiaj ntsig zoo hauv kev rho tawm modular thiab kev xam suav uas muaj cov lej loj.
4. Kev Ntsuas Hauv Paus:
Cov lus qhia no yog siv los sim cov hauv paus ntawm polynomials, uas yog lub hauv paus ntawm ntau cov lej algorithms hauv kev suav lej.
Cov Theorem seem ntawm Suav
Ntxiv rau qhov kev xav txog cov lej seem hauv cov ntsiab lus ntawm polynomials, kuj tseem muaj "Suav Cov Kev Xav Txog Cov lej seem" uas muaj kev siv dav hauv kev xav txog tus lej.
Xav tias peb muaj qee qhov kev sib npaug sib luag:
\[ x \equiv a_1 \ (\text{mod} \n_1) \]
\[ x \equiv a_2 \ (\text{mod} \n_2) \]
\[ \vdots \]
\[ x \equiv a_k \ (\text{mod} \n_k) \]
Qhov twg \(n_1, n_2, \ldots, n_k \) yog ib khub ntawm cov lej sib npaug ob npaug (ib khub ntawm cov lej uas tsis muaj cov yam ntxwv sib xws tsuas yog 1), Suav Remainder Theorem lav qhov muaj ib qho kev daws teeb meem tshwj xeeb modulo \(N \), qhov twg \(N \) yog cov khoom ntawm \(n_1, n_2, \ldots, n_k \).
Piv txwv ntawm kev siv Suav Remainder Theorem
Xav tias peb muaj lub kaw lus sib xws hauv qab no:
\[ x \equiv 2 \ (\text{mod} \ 3) \]
\[ x \equiv 3 \ (\text{mod} \ 5) \]
\[ x \equiv 2 \ (\text{mod} \ 7) \]
Peb yuav tsum nrhiav tus nqi x uas txaus siab rau tag nrho cov kab zauv no. Vim tias 3, 5, thiab 7 yog coprime, peb tuaj yeem siv Suav Remainder Theorem.
Kauj ruam thawj zaug yog xam \( N \):
\[ N = 3 \times 5 \times 7 = 105 \]
Kauj ruam thib ob yog xam \(N_i \) rau txhua moduli:
\[ N_1 = \frac{N}{3} = 35 \]
\[ N_2 = \frac{N}{5} = 21 \]
\[ N_3 = \frac{N}{7} = 15 \]
Kauj ruam thib peb yog nrhiav qhov sib npaug ntawm modulo \(N_i \) uas sib xws:
\[ 35x \equiv 1 \ (\text{mod} \ 3) \implies x = 2 \]
\[ 21x \equiv 1 \ (\text{mod} \ 5) \implies x = 1 \]
\[ 15x \equiv 1 \ (\text{mod} \ 7) \implies x = 1 \]
Tom qab ntawd muab nws tag nrho ua ke:
\[ x = a_1N_1x_1 + a_2N_2x_2 + a_3N_3x_3 \]
\[ x = 2 \cdot 35 \cdot 2 + 3 \cdot 21 \cdot 1 + 2 \cdot 15 \cdot 1 \]
\[ x = 140 + 63 + 30 = 233 \]
Thaum kawg, peb siv modulo N:
\[ x \equiv 233 \ (\text{mod} \ 105) \]
\[ x = 233 – 2 \cdot 105 \]
\[ x = 23 \]
Yog li ntawd, qhov kev daws teeb meem ntawm qhov system congruence yog \(x = 23 \).
Xaus
Cov theorem seem yog ib qho cuab yeej muaj zog thiab siv tau ntau yam hauv kev suav lej thiab lej. Yog tias koj nkag siab zoo, nws tuaj yeem ua kom cov kev suav nyuaj sai dua thiab qhib kev rau kev tshuaj xyuas ntxiv hauv kev suav lej. Nws cov ntawv thov suav nrog kev ntsuam xyuas polynomial, kev faib ua feem, cov lej integer, thiab kev daws cov kab ke sib luag, raws li pom hauv Suav Remainder Theorem. Los ntawm kev kawm cov theorem no, peb tuaj yeem txhim kho peb lub peev xwm los daws ntau yam teeb meem lej kom zoo dua thiab zoo dua.
