GORNER SXEMASINING MATEMATIK ANALIZGA
TATBIQI
Qaxxorova Muslimaxon Shokirjon qizi
Qabul qilingan: 03.04.2024
Qayta qabul : 10.04.2024
Saytda mavjud : 1.05.2024
Muallif (lar)
M.SH.Qaxxorova
Muallif bilan aloqa
https://orcid.org/0009-0004-3493-6680
muslimaxonqaxxorova@gmail.com
Muallif. M.SH.Qaxxorova
UNIVERSAL xalqaro ilmiy jurnal
https://universaljurnal.uz/index.php/jurnal
Maxfiylik bayonoti
Materialni istalgan vosita yoki formatda nusxalash va
qayta tarqatish hamda maqoladan
foydalanish mumkin.
ANNOTATSIYA
Barcha asosiy adabiyotlarda Gorner
sxemasi
x
c
k phad uchun berilgan.
K phadlarning ildizlarini topish juda muhim
ahamiyatga ega. Chunki k plab matematik
masalalarni yechish k phadning ildizlarini
rganish masalasiga olib keladi. Albatta, Gorner
sxemasi
bunda
muhim
ahamiyatga
ega.
Keltirilgan Gorner sxemasini
2
ax
bx c
kvadrat uchhad uchun sodda k rinishga
keltirilishi
k plab
masalalarni
yechishda,
teorema,
lemma,
xossalar
va
natijalarni
isbotlashda xizmat qiladi. Shuningdek undagi
umumlashtirishlar Gorner sxemasini q llash,
umumlashtirish b yicha ilmiy-amaliy xulosalar
chiqarish imkoniyatini beradi.
Ushbu maqolada kvadrat uchhad uchun
Gorner sxemasi keltirilgan b lib, undan
foydalanib k phadlarga va sodda kasrlarga
ajratishga doir masalalar yechib k rsatilgan.
Bundan tashqari ratsional kasrni sodda kasrga
ajratishda, matematik analizda ratsional kasr
k rinishdagi aniqmas integrallarni hisoblashni
osonlashtirish
maqsadida
foydalanish
mumkinligi k rsatilgan.
Kalit s zlar.
Gorner sxemasi, aniqmas
integral, ratsional kasr, ratsional funksiyalar,
integrallash.
Universal International Scientific Journal
2024, 1(1)
Universal Xalqaro Ilmiy Jurnal
Jurnalning bosh sahifasi:
https://universaljurnal.uz
Universal International Scientific Journal
2024, 1(1)
Annotation.
In all the main literature, the Gorner
scheme is given for the polynomial. Finding the
roots of polynomials is very important. Because
solving many mathematical problems leads to the
question of studying the roots of a polynomial. Of
course, the Gorner scheme is important in this.
Making the quoted Gorner scheme appear simple
for a square uchhad serves in solving many
problems, proving theorem, lemma, properties,
and results. Also, the information contained in it,
conclusions, theoretical generalizations provide
the opportunity to draw scientific and practical
conclusions on the application of the Gorner
scheme, generalization. This article presents the
Gorner scheme for a square uchhad, using which
the problems of decomposing into polynomials
and simple fractions are solved. Moreover, when
dividing a rational fraction by a simple fraction, it
is seen that in mathematical analysis the rational
fraction can be used with the aim of facilitating
the calculation of indeterminate integrals in the
representation.
Keywords.
Gorner's scheme, indeterminate integral,
rational fraction, rational functions, integrable.
KIRISH.
sib kelayotgan yosh avlodning bilimli,
intiluvchan, mustaqil fikrlaydigan va zamon
ruhiga mos shaxs sifatida tarbiyalash vazifasi
ustuvor masala b lib kelmoqda. Ushbu masalani
hal etish k
pedagogik
innovatsion
texnologiyalar,
zamonaviy axborot texnologiyalarni keng va
mukammal q
matematik tushunchani
qitilishida, q llashda,
avvalo, uning amaliy ahami
lozim.
Har
bir
matematik
tushunchani
qitilishida, q llashda, avvalo, uning amaliy
, mavjud
asosiy adabiyotlarda Gorner sxemasi
x
c
birhad uchun berilgan, unga doir amaliy
tatbiqlar
rganilgan.
Lekin
Gorner
sxemasini
2
ax
bx c
kvadrat uchhad
uchun deyarli
rganilmagan.
Yuqoridagi
holatlardan
kelib
chiqqan holda aytish mumkinki, Gorner
sxemasini
2
ax
bx c
kvadrat uchhad
uchun sodda k rinishga keltirish, uning
tatbiqlarini ilmiy tahlil etish, matematika
fani oldida turgan dolzarb muammolardan
biridir.
ADABIYOTLAR TAHLILI.
Adabiyotlar tahlili shuni k rsatdiki,
barcha asosiy adabiyotlarda Gorner sxemasi
x c
k phad
uchun
berilgan.
Ko`phadlarning ildizlarini topish juda
muhim ahamiyatga ega. Chunki ko`plab
matematik masalalarni yechish ko`phadning
ildizlarini o`rganish masalasiga olib keladi.
Albatta, Gorner sxemasi bunda muhim
ahamiyatga ega. Gorner sxemasi va uning
tatbiqlariga oid ko`plab olimlar ish olib
borgan. Jumladan, Ayupov Sh., Omirov B.,
quv q llanmasida
Gorner
sxemasi,
Bezu
teoremasi,
Algebraning asosiy teoremasi va shu kabilar
misollar yordamida aniq tushuntirilgan.
A.
Xojiev,
A.
Faynleybning
kitobida
kompleks koeffitsiyentli k phadlar haqida
tushuncha
berilgan.
Kompleks
koeffitsiyentli k phadlarning b linishi,
kompleks koeffitsiyentli k phadlarning
karrali ildizlari haqida tushunchalar hamda
kompleks koeffitsiyentli musbat darajali har
qanday k phad kompleks ildizga ega
ekanligi isbotlab berilgan. Bundan tashqari
berilgan.
Universal International Scientific Journal
2024, 1(1)
METODOLOGIYA.
Avvalo, Kvadrat uchhad uchun Gorner
sxemasini keltirib chiqaramiz. Aytaylik,
1
0
1
1
( )
...
n
n
n
n
f x
a x
a x
a
x
a
2
( )
h x
x
px
q
p
va
q
larning ishoralari qulaylik uchun tanlangan xolos.
( )
f x
va
( )
h x
darajasi
2
n
ga, qoldiqning darajasi birdan
2
3
0
1
3
2
( )
...
n
n
n
n
g x
b x
b x
b
x
b
va
( )
r x
cx
d
2
2
3
0
1
3
2
( )
...
n
n
n
n
f x
x
px q b x
b x
b x b
cx d
, quyidagi
tengliklar sistemasini hosil qilamiz.
0
0
a
b
1
0
1
a
b p
b
2
1
0
2
a
b p
b q
b
3
2
1
3
a
b p
b q
b
4
3
2
3
a
b p b q
b
...
1
2
3
n
n
n
a
b
p b q
c
2
n
n
a
b
q
d
Bu tengliklarni quyidagicha yozish mumkin:
0
0
b
a
1
0
1
b
b p
a
2
1
0
2
b
b p
b q
a
3
2
1
3
b
b p
b q
a
4
3
2
3
b
b p
b q
a
...
2
3
1
n
n
n
c
b
p
b q
a
2
n
n
d
b
q
a
Endi
xuddi
Gorner
sxemasiga
-
jadval):
1-jadval
0
a
1
a
2
a
...
1
n
a
n
a
p q
0
b
1
0
1
b
b p a
2
1
0
2
b
b p
b q a
2
3
1
n
n
n
c
b
p
b q a
2
n
n
d
b
q a
yuqoridagidek,
( )
f x
3
2
x
ax
bx c
ish mumkin.
Bu usulni qulaylik uchun Gornerning
ikkinchi sxemasi deb ataylik. Bu sxema yordamida
Bulardan
biri
2
n
m
P x
x
px
q
ratsional
kasrni sodda
kasrlarga ajratishdan iborat. Bu masalani ham
gorner
sxemasi
yordamida
yechishimiz
mumkin.
Ratsional funksiyalarni integrallash.
Bu keltirilgan Gornerning ikkinchi sxemasi
mumkin,
bunda
hisoblash
birmuncha
soddalashadi.
Universal International Scientific Journal
2024, 1(1)
NATIJALAR.
1-misol. Ushbu
5
3
3
2
3
3
2,
1
2
x
x
x
x
x
Yechish:
5
3
3
3
x
x
x
2
2
x
x
yozib olamiz.
1-jadvaldagi kabi koeffitsiyentlarni aniqlaymiz:
2-jadval
2
5
3
2
2
3
3 19 19
2
10 12
2
x
x x
x
x x
x
x x
5
3
3
2
3
3
19
19
6
8
x
x
x
x
x
x
x
2
2
2
2
2
2
2
2
10
12
19
19
2
2
10
12
2 19
19.
x
x
x
x
x
x
x
x
x
x
x
x
x
x
Endi tenglikning har ikkala tomonini hadma-had
3
2
2
x
x
5
3
3
2
3
2
2
2
2
3
3
2
10
22
19
19
;
2
2
2
2
x
x
x
x
x
2-misol. Ushbu
6
5
3
2
4
2
2
2
x
x
x
x
x
x
x
Yechish:
6
5
3
2
2
x
x
x
x
x
2
2
x
x
3-jadvaldagi
kabi
koeffitsiyentlarni
aniqlaymiz:
3-jadval
3
2
6
5
3
2
2
2
2
2
2
5
2
8
6
x
x
x
x
x
x
x
x
x
x
x
2
2
5
2
x
x
x
Endi tenglikning har ikkala tomonini hadma-
had
4
2
2
x
x
:
6
5
3
2
4
2
3
4
2
2
2
2
2
2
1
2 5
8 6
5 2
;
2
2
2
2
2
x x x x x
x x
3-misol. Ushbu
3
2
2
2
2
1
x
x
x
dx
x
x
aniqmas integralni hisoblang.
Yechish:
2-3-misoldagi
kabi
3
2
2
x
x
x
2
1
x
x
olamiz. 4-jadvaldagi kabi koeffitsiyentlarni
aniqlaymiz:
4-jadval
0
a
1
a
2
a
3
a
4
a
5
1
0
3
0
-1
3
1;2
1
1
6
8
19
19
1;2
1
2
10
12
0
a
1
a
2
a
3
a
4
a
5
6
a
1
1
0
1 -1
1
2
-1;-2
1
0
-2
3
0
-5
2
-1;-2
1
-1
-3
8
6
-1;-2
1
-2
-5
0
a
1
a
2
a
3
a
1
-1
1
2
-1;-1
1
-2
2
4
Universal International Scientific Journal
2024, 1(1)
3
2
2
2
2
1
2
4
x
x
x
x
x
x
x
Endi tenglikning har ikkala tomonini
hadma-had
2
2
1
x
x
:
3
2
2
2
2
2
2
2
2
2
4
1
1
1
x
x
x
x
dx
dx
x
x
2
2
2
2
2
2
2
4
2
1
ln
1
1
1 2
1
x
dx
dx
1
2
1
3
3
x
arctg
C
Bu keltirilgan Gornerning ikkinchi sxemasi
yordamida bundan boshqa ko`plab masalalarni
yechish uchun ham qo`llash mumkin, bu esa
hisoblashni yengillashtirishga yordam beradi.
MUHOKAMA.
Ilmiy
masalalar
shundan
iboratki,
Gorner
sxemasining kvadrat uchhad uchun formulasi ishlab
chiqilgan va u yordamida ratsional funksiyali
aniqmas
integrallarni
osongina
hisoblashda
funksiyalarni osonlashtirish ahamiyatli hisoblanadi,
yengillashtirish mumkin.
x c
birhad
lalar yechiladi.
Bunday masalalarga
f x
x c
f x
x c
f x
x c
birhadning
darajalari
f x
x c
birhad karrali ildiz ekanligini aniqlash va h.k.
XULOSA.
Ushbu maqolada butun koeffitsiyentli
Gorner sxemasini umumlashtirish va uning
tadbiqlari
haqida
b lib,
unda
Gorner
sxemasini
2
ax
bx c
kvadrat uchhad
uchun sodda k rnishga keltirish uning
tatbiqlarini
ochib
berish
masalalariga
qaratilgan b lib, bu b yicha dastlab olib
borilgan
tadqiqotlar
natijasida
quyidagi
xulosalar taqdim etildi:
1.
1
1
1
0
...
n
n
n
n
P x
a x
a
x
a x
a
k phadni
2
ax
bx c
kvadrat uchhadga
b lgandagi qoldiqni topishga q llangan;
2.
1
1
1
0
...
n
n
n
n
P x
a x
a
x
a x
a
k phadni
2
ax
bx c
kvadrat uchhadning
darajalari b yicha yoyilgan
3.
2
ax
bx c
kvadrat uchhad
uchun Gorner sxemasini
2
n
m
P x
x
px
q
k rinishdagi ratsional kasrni sodda kasrlarga
ajratish mumkinligi k rsatilgan;
4.
2
ax
bx c
kvadrat uchhad
uchun
Gorner
sxemasini
q llab
Universal International Scientific Journal
2024, 1(1)
2
n
m
P x
dx
x
px
q
k rinishdagi
aniqmas
integrallar k rsatilgan.
Bu
esa
Gorner
sxemasining
matematik analizga tatbiqi b yicha dastlabki
qadam b lib xizmat qiladi.
FOYDALANILGAN ADABIYOTLAR
1. Abdulxamidov A.,
1-
-2008. B: 121-141.
masalalar t
-qism, To
-2003, B:57-63.
3.
.
h
-2011. B:125-
128.
4.
quv
q
ur b
-2019. B: 95-128.
5.
Raxmonova, V. (2023). THE ROLE AND PLACE OF MATHEMATICAL MODELS IN
TEACHING STUDENTS TO SOLVE OPTIMIZATION PROBLEMS.
Modern Science and
Research
,
2
(4), 592-597.
