+36 ko’phadning ildizlarini toping. 2-misol f(x)=2x

1-misol f(x)=x⁴-13x²+36 ko’phadning ildizlarini toping.

2-misol. f(x)=2x⁵+x⁴-10x³-5x²+8x+4=0 ko’phadning ildizlarini toping.

3-misol. Kasrlarni qisqartiring:

a)  b) 

c)  d) 

4-misol.  ifodani soddalashtiring.

5-misol.  ifodani soddalashtiring.

6-misol . tenglamaning ildizlari ko’paytmasini toping.

7-misol. tenglamani ildizlari yig’indisini toping.

8- misol. x³+2x²-9x-18=0 tenglamani ildizlari yig’indisini toping.

9. Ushbu Ushbu x³-px²-qx+4=0 tenglamaning ildizlaridan biri 1 ga teng.Shu

tenglama barcha koeffitsientlarini yig‘indisini toping.

10. tenglamani yeching
1. Ifodani soddalashtiring.

a) b)

d)

2. Agar bo’lsa x²(4-x) ning qiymatini toping.

3. =0 tenglamaning eng katta yechimini toping.

4. Tenglamalarni yeching.

a) 3x⁴-48=0 ; b) x⁶-27=0 ;

c) 12x⁴-8x=0 ; d)x⁶-16x²=0 ;

e) x³-64=0 ; f) x²-49=0 ;

g) 2x⁶-54=0 h) 5x³-6x=0 ;

5-misol. x⁴ +4x³-10x²-28x-15=0 tenglamani yeching.

6. tenglamaning ildizlaridan biri -2 ga teng . uning ikkinchi ildizini toping.

7. (x²-5)²+x²-5=0 tenglamanig ildizlari ko’paytmasini toping.

8. Qaytma tenglamalarni yeching.

3x⁴+2x³-4x²+2x+3=0 ;

2x⁴-4x³+8x²-4x+2=0

21x⁶+82x⁵+103x⁴+164x³+103x²+82x+21=0
1. P(x) ni Q(x) ga bo’lgandagi qoldiqni toping. Bo’lishni burchak usuli yordamida tekshiring.

P(x) = 32х^4-64х^3 + 8х^2 + 36х + 4 Q(x) = 2х-1

P(x) = х^4-4х^3 + 7х^2-12х + 12 Q(x) = х-2

2. P(x)= x + x³ + x⁹ + x²⁷ + x⁸¹ + x²⁴³ ko’phadni  Q(x)=  x – 1 ko’phadga bo’lgandagi qoldiqni toping

3. P(x) = (x + 1)⁶ – x⁶ – 2x – 1  ko’phad Q(x)= x(x + 1)(2x + 1) ko’phadga bo’linishini ko’rsating.

4. a va b ning qanday qiymatlarida  P(x) = (a + b)x⁵ + abx² + 1 ko’phad Q(x)= x² – 3x + 2 ko’phdga qoldiqsiz bo’linadi?

5. a ning qanday qiymatlarida  P(x) = xⁿ + ax^n–2  (n ≥ 2) ko’phad Q(x)= x – 2 ko’phdga qoldiqsiz bo’linadi?

6. a ning qanday qiymatlarida  P(x) = a³x⁵ + (1 – a)x⁴ + (1 + a³)x² + (1 – 3a)x – a³  ko’phad Q(x)= x – 1 ko’phdga qoldiqsiz bo’linadi?

7. P(x)=xⁿ + x + 2 ko’phadni Q(x)= x² – 1ko’phadga bo’lgandagi R(x) qoldiqni toping.

8. P(x) = x⁵ – 17x + 1  ko’phadni Q(x)= на  x + 2 ko’phadga bo’lgandagi R(x) qoldiqni toping.

9. P(x) = x⁸¹ + x²⁷ + x⁹ + x³ + x  ko’phadni quyida berilgan ko’phadlarga bo’lgandagi qoldiqni toping.  a)  x – 1;    b)  x² – 1.

10. a³(b² – c²) + b³(c² – a²) + c³(a² – b²) ko’phadni  (b – c)(c – a)(a – b)ko’phadga bo’linishini isbotlang.
1. P(x) ni Q(x) ga bo’lgandagi qoldiqni toping. Bo’lishni burchak usuli yordamida tekshiring.

P(x) = 32х^4-64х^3 + 8х^2 + 36х + 4 Q(x) = 2х-1

P(x) = х^4-4х^3 + 7х^2-12х + 12 Q(x) = х-2

2. P(x)= x + x³ + x⁹ + x²⁷ + x⁸¹ + x²⁴³ ko’phadni  Q(x)=  x – 1 ko’phadga bo’lgandagi qoldiqni toping

3. P(x) = (x + 1)⁶ – x⁶ – 2x – 1  ko’phad Q(x)= x(x + 1)(2x + 1) ko’phadga bo’linishini ko’rsating.

4. a va b ning qanday qiymatlarida  P(x) = (a + b)x⁵ + abx² + 1 ko’phad Q(x)= x² – 3x + 2 ko’phdga qoldiqsiz bo’linadi?

5. a ning qanday qiymatlarida  P(x) = xⁿ + ax^n–2  (n ≥ 2) ko’phad Q(x)= x – 2 ko’phdga qoldiqsiz bo’linadi?

6. a ning qanday qiymatlarida  P(x) = a³x⁵ + (1 – a)x⁴ + (1 + a³)x² + (1 – 3a)x – a³  ko’phad Q(x)= x – 1 ko’phdga qoldiqsiz bo’linadi?

7. P(x)=xⁿ + x + 2 ko’phadni Q(x)= x² – 1ko’phadga bo’lgandagi R(x) qoldiqni toping.

8. P(x) = x⁵ – 17x + 1  ko’phadni Q(x)= на  x + 2 ko’phadga bo’lgandagi R(x) qoldiqni toping.

9. P(x) = x⁸¹ + x²⁷ + x⁹ + x³ + x  ko’phadni quyida berilgan ko’phadlarga bo’lgandagi qoldiqni toping.  a)  x – 1;    b)  x² – 1.

10. a³(b² – c²) + b³(c² – a²) + c³(a² – b²) ko’phadni  (b – c)(c – a)(a – b)ko’phadga bo’linishini isbotlang.

1. Gorner sxamasi yordamida P(x)=5x⁴ +5x³ +x² −11 ko’phadni Q(x)= x−1 ko’phadga bo’ling.

2. Gorner sxamasi yordamida P(x)= x⁴+3x³+4x²−5x−47 ko’phadni Q(x)= x+3 ko’phadga bo’ling.

3. P(x)= x⁶+2x⁵−21x⁴−20x³+71x²+114x+45 Gorner sxemasi yordamida ko’phadning barcha butun yechimlarini toping

4. P(x)=3x⁶+9x⁵−28x⁴+6x³−30x²−30x+100 ko’phadni 2 va -5 sonlar ildizibo’lishini tekshiring. Berilgan ko’phadni x-2 va x+5 ko’phadlarga bo’ling.

5. 2x³-11x²+12x+9 ko’phadni ikkihadga bo’linishini tekshiring.

6. x³-7x-6=0 tenglama ildizlarini toping va tenglikni chap qismini ko’paytuvchilarga ajrating.

7. Gorner sxamasi yordamida P(x) ni Q(x) ga bo’ling.

P(x) = х³ + 3х²-18х-40 Q(x) = х + 2

P(x) = х⁴ + 2х³-3х² + 5х-2 Q(x) = 2х-3

8. Ko’phadni ikkihadga bo’lgandagi qoldiqni toping.

a) 2x³-5x²+3x+7 ni x-2 ga

b) x⁴+x³+x²-2x+4 ni x+3 ga d) x²+5x-6 ni x-2 ga

9. Berilgan ko’phadni ikkihadga bo’linishini qoldiqsiz bo’linishini ko’rsating.

a) 2x⁴-3x³-7x²+6x+8 va x-2  b) 2x³-5x²+1 ni x+1/2 ga

10. Gorner sxamasi yordamida P(x) ni Q(x) ga bo’ling.

P(x)=2x⁴-x³-9x²+13x-5 va Q(x)=x-2 

11. Gorner sxamasi yordamida P(x) ni Q(x) ga bo’lgandagi qoldiqni toping.

P(x)= x⁶−4x⁴+x³−2x²+5 Q(x)=x+3 

1. Gorner sxamasi yordamida P(x)=5x⁴ +5x³ +x² −11 ko’phadni Q(x)= x−1 ko’phadga bo’ling.

2. Gorner sxamasi yordamida P(x)= x⁴+3x³+4x²−5x−47 ko’phadni Q(x)= x+3 ko’phadga bo’ling.

3. P(x)= x⁶+2x⁵−21x⁴−20x³+71x²+114x+45 Gorner sxemasi yordamida ko’phadning barcha butun yechimlarini toping

4. P(x)=3x⁶+9x⁵−28x⁴+6x³−30x²−30x+100 ko’phadni 2 va -5 sonlar ildizibo’lishini tekshiring. Berilgan ko’phadni x-2 va x+5 ko’phadlarga bo’ling.

5. 2x³-11x²+12x+9 ko’phadni ikkihadga bo’linishini tekshiring.

6. x³-7x-6=0 tenglama ildizlarini toping va tenglikni chap qismini ko’paytuvchilarga ajrating.

7. Gorner sxamasi yordamida P(x) ni Q(x) ga bo’ling.

P(x) = х³ + 3х²-18х-40 Q(x) = х + 2

P(x) = х⁴ + 2х³-3х² + 5х-2 Q(x) = 2х-3

8. Ko’phadni ikkihadga bo’lgandagi qoldiqni toping.

a) 2x³-5x²+3x+7 ni x-2 ga

b) x⁴+x³+x²-2x+4 ni x+3 ga d) x²+5x-6 ni x-2 ga

9. Berilgan ko’phadni ikkihadga bo’linishini qoldiqsiz bo’linishini ko’rsating.

a) 2x⁴-3x³-7x²+6x+8 va x-2  b) 2x³-5x²+1 ni x+1/2 ga

10. Gorner sxamasi yordamida P(x) ni Q(x) ga bo’ling.

P(x)=2x⁴-x³-9x²+13x-5 va Q(x)=x-2 

11. Gorner sxamasi yordamida P(x) ni Q(x) ga bo’lgandagi qoldiqni toping.

P(x)= x⁶−4x⁴+x³−2x²+5 Q(x)=x+3 

.

1. Gorner sxemasi yordamida tenglamalarni yeching.

a) x⁴+7x²-24=0 ; b) x³+4x-80=0 ; d) 7x⁴-5x³+2x=0 ;

f) 9x⁴+4x³-13x²=0 ; g) x⁴-8x²-9=0

h) x⁴-x³-x²+x-5=0 ; i)x⁵+x⁴+3x=0 ; j) 6x²+7x-81=0 ;

k) x⁴+5x³-7x²-x-5=0 ; l) x³+3x²-4x-6=0 ; ; m) 6x²+5x-90=0 ;

2-misol f(x)=x⁴-13x²+36 ko’phadning ildizlarini toping.

3-misol. f(x)=2x⁵+x⁴-10x³-5x²+8x+4=0 ko’phadning ildizlarini toping.

4-misol. tenglamani yeching.

5-misol. x³+4x²-3x+5 ko’phadni Gorner sxemasidan foydalanib, x-1 ga bo’lishni bajaring.

6-misol. P(x)= x³-3x²+5x+7 ni 2x+1 ga bo’lishdan hosil bo’lgan qoldiqni toping.

7-misol. P₄(x) = x⁴+x³+3x²+2x+2 ko’phadni x-1 ga bo’lishdan hosil bo’lgan qoldiqni toping

8-misol: P₅(x)= 2x⁵ –x⁴-3x³+x-3 ni x-3 ga bo’lishdan hosil bo’lgan qoldiqni toping.

9-misol. F(x)=2x⁵+x⁴-10x³-5x²+8x+4 ko’phadning ildizlarini toping.

10-misol. F(x)=x⁴-13x²+36 ko’phadning ildizlarini toping.

11-misol. Gorner sxemasidan foydalanib, f(x) ko’phadning x=a nuqtadagi qiymatini toping.

1) f(x)= ; 2) f(x)= ; 3) f(x)=
1. Gorner sxemasi yordamida tenglamalarni yeching.

a) x⁴+7x²-24=0 ; b) x³+4x-80=0 ; d) 7x⁴-5x³+2x=0 ;

f) 9x⁴+4x³-13x²=0 ; g) x⁴-8x²-9=0

h) x⁴-x³-x²+x-5=0 ; i)x⁵+x⁴+3x=0 ; j) 6x²+7x-81=0 ;

k) x⁴+5x³-7x²-x-5=0 ; l) x³+3x²-4x-6=0 ; ; m) 6x²+5x-90=0 ;

2-misol f(x)=x⁴-13x²+36 ko’phadning ildizlarini toping.

3-misol. f(x)=2x⁵+x⁴-10x³-5x²+8x+4=0 ko’phadning ildizlarini toping.

4-misol. tenglamani yeching.

5-misol. x³+4x²-3x+5 ko’phadni Gorner sxemasidan foydalanib, x-1 ga bo’lishni bajaring.

6-misol. P(x)= x³-3x²+5x+7 ni 2x+1 ga bo’lishdan hosil bo’lgan qoldiqni toping.

7-misol. P₄(x) = x⁴+x³+3x²+2x+2 ko’phadni x-1 ga bo’lishdan hosil bo’lgan qoldiqni toping

8-misol: P₅(x)= 2x⁵ –x⁴-3x³+x-3 ni x-3 ga bo’lishdan hosil bo’lgan qoldiqni toping.

9-misol. F(x)=2x⁵+x⁴-10x³-5x²+8x+4 ko’phadning ildizlarini toping.

10-misol. F(x)=x⁴-13x²+36 ko’phadning ildizlarini toping.

11-misol. Gorner sxemasidan foydalanib, f(x) ko’phadning x=a nuqtadagi qiymatini toping.

1) f(x)= ; 2) f(x)= ; 3) f(x)=

x²+y²=(x+y)²-2xy

x³+y³=(x+y)³-3xy(x+y)

x⁴+y⁴= 2xy(x²+y²)-(x⁴+y⁴)+3(xy)² и т.д.

Теорема 4. Элемент a поля K является корнем многочлена f(x) ненулевой степени (то есть f(a) = 0) тогда и только тогда, когда f(x) = (x − a) · p(x) для некоторого многочлена p(x)

6-misol. 3x⁴+3x³+4x²-50=0

7-misol. x⁴+x³-8x²-2x+8=0 ;

4-misol. tenglamalarni yeching.

a) x⁴+7x²-24=0 ; b) x³+4x-80=0 ; d) 7x⁴-5x³+2x=0 ;

e) x²+4x+4=0 ; f) 9x⁴+4x³-13x²=0 ; g) x⁴-8x²-9=0

h) x⁴-x³-x²+x-5=0 ; i)x⁵+x⁴+3x=0 ; j) 6x²+7x-81=0 ;

k) x⁴+5x³-7x²-x-5=0 ; l) x³+3x²-4x-6=0 ; ; m) 6x²+5x-90=0 ;

n) 5x⁵-4x⁴+3x³-2x²+x-3=0 ;