Algebra

Viyet teoremasi

murakkab 185 daqiqa Viyet teoremasiVieta theoremsum of rootsproduct of rootsinverse Vietasymmetric expressionstransformed rootsreciprocal rootsconjugate rootscubic VietaNewton sumsroot coefficients

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Viyet teoremasi ildizlarni alohida hisoblamasdan ular haqidagi katta miqdordagi ma’lumotni koeffitsientlardan olish imkonini beradi. Bu parametrli masalalar, yangi tenglama tuzish, symmetric ifodalarni hisoblash, integer-root savollari va yuqori darajali polynomiallarga o‘tish uchun eng kuchli algebraik ko‘priklardan biridir. Asosiy maqsad formula yodlash emas, ildizlar o‘rniga ularning symmetric yig‘indi va ko‘paytmalarini ishlatish strategiyasini shakllantirishdir.

O'quv maqsadlari

ax²+bx+c=0 tenglamani monic ko‘rinishga keltirish
Kvadrat Viyet formulalari x₁+x₂=−b/a va x₁x₂=c/a ni qo‘llash
Ildizlarni topmasdan symmetric expressionlarni hisoblash
x₁²+x₂² ni sum/product orqali chiqarish
x₁³+x₂³ ni sum/product orqali chiqarish
1/x₁+1/x₂ kabi reciprocal expressionsni hisoblash
x₁/x₂+x₂/x₁ ni symmetric shaklga keltirish
x₁²x₂+x₂²x₁ ni factorlash
Shifted reciprocal expressionsni sum/product orqali hisoblash
Berilgan sum va productdan monic quadratic tuzish
Berilgan ikki ildizdan quadratic equation tuzish
Original rootsga teskari sonlar uchun yangi tenglama tuzish
Qarama-qarshi roots uchun yangi tenglama tuzish
Rootsni k ga siljitish va k marta masshtablashdan yangi equation tuzish
Ratsional koeffitsientli polynomialda quadratic irrational conjugate rootni aniqlash
Rootlardan aralash transformatsiya qilingan yangi roots uchun equation tuzish
Cubic Vieta formulalarini sum, pairwise product va product uchun qo‘llash
Cubic roots power sums uchun Newton identity bazasini ishlatish
Symmetric va symmetric bo‘lmagan root ifodalarni farqlash
Savol bankidagi noto‘g‘ri marked answer yoki buzilgan source matnni independent algebraik QA bilan ajratish

Kvadrat tenglama ildizlarini topish shart bo‘lmagan savollar juda ko‘p. Agar ax²+bx+c=0 ning ildizlari x₁ va x₂ bo‘lsa, factorization a(x−x₁)(x−x₂) ni ochish orqali x₁+x₂=−b/a va x₁x₂=c/a kelib chiqadi. Shu ikki son S=x₁+x₂ va P=x₁x₂ ko‘plab symmetric ifodalarni boshqaradi. Masalan x₁²+x₂²=S²−2P, x₁³+x₂³=S³−3PS, 1/x₁+1/x₂=S/P. Teskari Viyet esa aksincha ishlaydi: sum S va product P ma’lum bo‘lsa, rootsi shu sonlar bo‘lgan monic equation x²−Sx+P=0. Bankning yuqori qatlamida rootsni teskari, qarama-qarshi, k ga katta yoki k marta katta qilish, quadratic irrational conjugates va cubic Vieta/Newton sums mavjud. Bular yadro Viyetning bir xil g‘oyasini davom ettiradi: individual roots o‘rniga symmetric invariants bilan ishlash.

Ta'riflar

Viyet teoremasi · Viète's formulas · Теорема Виета

Polynomial koeffitsientlarini uning ildizlarining elementary symmetric yig‘indilari bilan bog‘lovchi formulalar.

Ildizlarni yechmasdan ularning sum/productini koeffitsientlardan topish.

Misol: ax²+bx+c=0 da x₁+x₂=−b/a, x₁x₂=c/a.

Bu emas: x₁=quadratic formula bilan individual root topish Viyetning o‘zi emas.

💡 Kvadrat holat bu umumiy polynomial teoremaning eng ko‘p ishlatiladigan ko‘rinishi.

Ildizlar yig‘indisi · Sum of roots · Сумма корней

S=x₁+x₂.

Ikki ildizning symmetric yig‘indisi.

Misol: x²−7x+12=0 da S=7.

Bu emas: x₁−x₂ symmetric sum emas.

💡 Quadraticda S=−b/a.

Ildizlar ko‘paytmasi · Product of roots · Произведение корней

P=x₁x₂.

Ikki ildizning symmetric producti.

Misol: x²−7x+12=0 da P=12.

Bu emas: x₁/x₂ product emas.

💡 Quadraticda P=c/a.

Keltirilgan quadratic · Monic quadratic · Приведённое квадратное уравнение

x²+px+q=0, ya’ni leading coefficient 1 bo‘lgan quadratic.

Viyet belgilarini eng sodda ko‘rsatadigan form.

Misol: x²−5x+6=0.

Bu emas: 2x²−5x+6=0 monic emas.

💡 a≠0 bo‘lsa barcha hadni a ga bo‘lib monic qilish mumkin.

Teskari Viyet · Inverse Vieta · Обратная теорема Виета

Sum S va product P berilgan bo‘lsa, shu ikki son rootsi bo‘lgan monic quadratic x²−Sx+P=0.

Roots xossalaridan equationni qayta qurish.

Misol: Roots 2 va 5 → x²−7x+10=0.

Bu emas: x²+7x+10=0 roots 2 va 5 emas.

💡 Real roots mavjudligi kerak bo‘lsa D=S²−4P≥0.

Symmetric ifoda · Symmetric expression · Симметрическое выражение

x₁ va x₂ o‘rin almashtirganda qiymati o‘zgarmaydigan ifoda.

Root orderingdan mustaqil ifoda.

Misol: x₁²+x₂², x₁x₂, x₁/x₂+x₂/x₁.

Bu emas: x₁−x₂ o‘rin almashtirganda ishora almashadi.

💡 Viyet to‘g‘ridan-to‘g‘ri symmetric expressions uchun tabiiy.

Elementary symmetric sums · Elementary symmetric functions · Элементарные симметрические суммы

Rootsning 1-darajali sum, pairwise product sum, ... va umumiy producti.

Polynomial koeffitsientlari bilan bevosita bog‘langan asosiy symmetric quantities.

Misol: Cubicda e₁=r₁+r₂+r₃, e₂=r₁r₂+r₂r₃+r₃r₁, e₃=r₁r₂r₃.

Bu emas: r₁²+r₂²+r₃² elementary symmetric emas.

💡 General Vieta shu eₖ larni beradi.

Power sum · Power sum · Сумма степеней корней

p_n=x₁^n+x₂^n (yoki barcha roots uchun mos sum).

Rootsning bir xil darajadagi yig‘indisi.

Misol: p₃=x₁³+x₂³.

Bu emas: x₁²x₂ mixed term power sum emas.

💡 Newton identities bilan recursive topilishi mumkin.

Newton identiklari · Newton identities · Тождества Ньютона

Power sumsni elementary symmetric sums, demak polynomial koeffitsientlari bilan bog‘lovchi recurrence formulalar.

Yuqori darajali root powerlarni rootsni topmasdan hisoblash vositasi.

Misol: Quadraticda p₂=S²−2P, p₃=S³−3PS.

Bu emas: Har qanday nonsymmetric ifodani avtomatik hisoblab bermaydi.

💡 Bank-extension uchun ishlatiladi.

Teskari ildizlar · Reciprocal roots · Обратные корни

P≠0 bo‘lganda 1/x₁ va 1/x₂.

Har rootni reciprocal qilish.

Misol: x²−10x+21 roots reciprocal equation: 21y²−10y+1=0.

Bu emas: xᵢ=0 bo‘lsa reciprocal aniqlanmagan.

💡 New sum S/P, new product 1/P.

Qarama-qarshi ildizlar · Opposite roots · Противоположные корни

−x₁ va −x₂.

Har root ishorasini almashtirish.

Misol: x²−15x+50 → opposite roots equation x²+15x+50=0.

Bu emas: Reciprocal roots boshqa transformatsiya.

💡 Sum signi almashadi, product o‘zgarmaydi.

Siljitilgan ildizlar · Shifted roots · Сдвинутые корни

x₁+k va x₂+k.

Har rootga bir xil k qo‘shish.

Misol: Roots 3,4; +5 → 8,9.

Bu emas: kx₁ scaling shift emas.

💡 New S=S+2k, P=P+kS+k².

Masshtablangan ildizlar · Scaled roots · Масштабированные корни

kx₁ va kx₂.

Har rootni bir xil k ga ko‘paytirish.

Misol: Roots 3,5; twice →6,10.

Bu emas: xᵢ+k shift boshqa.

💡 New S=kS, P=k²P.

Quadratic conjugate roots · Quadratic conjugates · Сопряжённые квадратичные иррациональности

Ratsional koeffitsientli quadraticda p+q√d root bo‘lsa, p−q√d ham root bo‘ladigan conjugate juftlik.

Irrational qismning ishorasi almashgan juft.

Misol: 1+√2 va 1−√2.

Bu emas: 1+√2 va −1+√2 conjugate emas.

💡 Sum va product rational bo‘lib equation koeffitsientlari rational qoladi.

Cubic Viyet · Cubic Vieta · Формулы Виета для кубического

ax³+bx²+cx+d=0 roots r₁,r₂,r₃ uchun e₁=−b/a, e₂=c/a, e₃=−d/a.

Uch rootsning sum, pairwise-product sum va productini koeffitsientlardan topish.

Misol: x³−2x²−2x+1: e₁=2,e₂=−2,e₃=−1.

Bu emas: e₂ faqat r₁r₂ emas; uchta pair yig‘indisi.

💡 Bankdagi cubic extensionning asosi.

Root transform · Root transformation · Преобразование корней

Original rootsga bir xil funksiya qo‘llab yangi roots yᵢ=f(xᵢ) hosil qilish.

Yangi equation uchun new sum/productni hisoblash.

Misol: yᵢ=2xᵢ yoki yᵢ=xᵢ+5.

Bu emas: x₁ ga bitta, x₂ ga boshqa unrelated rule berish symmetric transform emas.

💡 Linear, reciprocal va mixed transforms bankda ko‘p.

Root invariant · Symmetric invariant · Симметрический инвариант

Root ordering o‘zgarsa ham bir xil qoladigan quantity.

Viyet orqali koeffitsientlardan aniqlanadigan xususiyat.

Misol: S, P, S²−2P.

Bu emas: x₁−x₂ signi orderingga bog‘liq.

💡 Canonical QA’da aynan shu farq muhim.

Rational-coefficient condition · Rational coefficient condition · Условие рациональных коэффициентов

Equation koeffitsientlari rational bo‘lishi rootlarning symmetric sum/producti rational bo‘lishini talab qiladi.

Irrational rootning sherigi ko‘pincha conjugate orqali irrational qismlarni bekor qiladi.

Misol: Root −2+3√2 bo‘lsa conjugate −2−3√2.

Bu emas: −2−2√2 ni avtomatik ikkinchi root deb olish noto‘g‘ri.

💡 Quadratic irrational minimal polynomial tuzishda ishlaydi.

Fundamental tushunchalar

Monic normalization

General ax²+bx+c=0 ni a ga bo‘lib x²+(b/a)x+c/a=0 qilinsa Viyet belgilari ko‘rinadi.

$x^2+\frac ba x+\frac ca=0$

a≠0.

Quadratic Vieta yadro formulasi

Roots sum coefficientning qarama-qarshi ratio’si, product esa constant/leading ratio.

$S=-b/a,\quad P=c/a$

Rootlarni individual topish shart emas.

Factor expansion orqali sabab

a(x−x₁)(x−x₂) ochilsa middle coefficient −a(x₁+x₂), constant ax₁x₂ chiqadi.

$a[x^2-Sx+P]$

Coefficient matching Viyetni beradi.

Inverse Vieta

Sum S va product P ma’lum bo‘lsa roots shu bo‘lgan monic equation x²−Sx+P=0.

$x^2-Sx+P=0$

Equation constructionning asosiy modeli.

Symmetry first

Ifoda x₁↔x₂ almashganda o‘zgarmasa, uni S va P orqali yozishga harakat qiling.

$f(x_1,x_2)=f(x_2,x_1)$

Viyetning strategik gate’i.

Square sum identity

x₁²+x₂²=(x₁+x₂)²−2x₁x₂.

$x_1^2+x_2^2=S^2-2P$

Eng ko‘p ishlatiladigan symmetric transform.

Cube sum identity

x₁³+x₂³=(x₁+x₂)³−3x₁x₂(x₁+x₂).

$x_1^3+x_2^3=S^3-3PS$

Rootsni explicit topmasdan cube sum.

Mixed cubic factor

x₁²x₂+x₂²x₁=x₁x₂(x₁+x₂).

$P S$

Mixed terms ham symmetric bo‘lsa S,P ga tushadi.

Reciprocal sum

P≠0 bo‘lsa 1/x₁+1/x₂=(x₁+x₂)/(x₁x₂).

$S/P$

c≠0 kerak.

Ratio sum

x₁/x₂+x₂/x₁=(x₁²+x₂²)/P.

$(S^2-2P)/P$

P≠0.

Shifted reciprocal

1/(x₁+k)+1/(x₂+k) numerator va denominatorni S,P orqali beradi.

$\frac{S+2k}{P+kS+k^2}$

Denominator nonzero.

Linear-combination denominators

1/(x₁+mx₂)+1/(x₂+mx₁) symmetric qilib expansion qilinadi.

$\frac{(1+m)S}{mS^2+(m-1)^2P}$

Denominators nonzero.

Root equation reconstruction

Yangi roots y₁,y₂ uchun faqat S_y va P_y ni topish kifoya.

$y^2-S_y y+P_y=0$

Leading coefficientni xohlagan nonzero scale bilan ko‘paytirish mumkin.

Reciprocal-root equation

Original ax²+bx+c=0, c≠0 bo‘lsa reciprocal roots uchun cx²+bx+a=0.

$c y^2+b y+a=0$

Coefficient order tashqi ikki hadlarda almashadi.

Opposite-root equation

x→−y substitution bilan odd-degree coefficientlar ishorasi almashadi.

$ay^2-by+c=0$

Quadraticda b signi almashadi.

Shift transform

Roots xᵢ+k uchun sum/product formulasidan yoki original polynomialda x=y−k substitutiondan foydalanish mumkin.

$S_k=S+2k,\ P_k=P+kS+k^2$

Ikki usul bir xil.

Scale transform

Roots kxᵢ bo‘lsa sum kS, product k²P.

$S_k=kS,\ P_k=k^2P$

k=0 bo‘lsa ikkala yangi root 0.

Quadratic irrational conjugates

Rational coefficients uchun p+q√d rootning conjugate p−q√d sherigi sum/productni rational qiladi.

$(x-(p+q\sqrt d))(x-(p-q\sqrt d))$

=x²−2px+(p²−q²d).

Cubic Vieta

Cubic roots uchun uchta elementary symmetric quantity koeffitsientlardan olinadi.

$e_1=-b/a,\ e_2=c/a,\ e_3=-d/a$

Root orderingdan mustaqil.

Cubic square sum

r₁²+r₂²+r₃²=e₁²−2e₂.

$p_2=e_1^2-2e_2$

Quadratic identityning 3-root analogi.

Newton power recurrence

Power sums pₖ elementary symmetric coefficientsdan recurrence orqali topiladi.

$p_1=e_1,\ p_2=e_1p_1-2e_2,\ p_3=e_1p_2-e_2p_1+3e_3$

Cubic extensionda ishlatiladi.

Nonsymmetric QA gate

Root ordering berilmasa x₁−x₂ yoki unequal-weight pair expression kabi nonsymmetric quantity odatda koeffitsientlardan yagona aniqlanmaydi.

$x_1\leftrightarrow x_2$

Almashganda o‘zgarsa ordering kerak yoki absolute/square bilan symmetric qilish lozim.

Formula kutubxonasi

Quadratic root sum

$$x_1+x_2=-\frac ba$$

Ildizlar yig‘indisini koeffitsientlardan beradi.

Shart: ax²+bx+c=0, a≠0

Quadratic root product

$$x_1x_2=\frac ca$$

Ildizlar ko‘paytmasi.

Shart: a≠0

Inverse Vieta equation

$$x^2-Sx+P=0$$

Berilgan roots xossalaridan monic equation tuzadi.

Shart: Roots sum S, product P

Square sum

$$x_1^2+x_2^2=S^2-2P$$

Root squares sum.

Cube sum

$$x_1^3+x_2^3=S^3-3PS$$

Root cubes sum.

Mixed cubic

$$x_1^2x_2+x_2^2x_1=PS$$

Mixed symmetric degree-3 expression.

Reciprocal sum

$$\frac1{x_1}+\frac1{x_2}=\frac SP$$

Reciprocal roots sum.

Shart: P≠0

Reciprocal product

$$\frac1{x_1x_2}=\frac1P$$

Reciprocal roots product.

Shart: P≠0

Ratio sum

$$\frac{x_1}{x_2}+\frac{x_2}{x_1}=\frac{S^2-2P}{P}$$

Symmetric ratio sum.

Shart: P≠0

Shifted reciprocal sum

$$\frac1{x_1+k}+\frac1{x_2+k}=\frac{S+2k}{P+kS+k^2}$$

Shifted denominator expression.

Shart: (x₁+k)(x₂+k)≠0

Linear-combination reciprocal sum

$$\frac1{x_1+mx_2}+\frac1{x_2+mx_1}=\frac{(1+m)S}{mS^2+(m-1)^2P}$$

Cross-mixed root denominatorsni S,P ga tushiradi.

Shart: Denominators nonzero

Opposite-root equation

$$ay^2-by+c=0$$

Qarama-qarshi roots equation.

Shart: Original ax²+bx+c=0 roots xᵢ; new roots yᵢ=−xᵢ

Reciprocal-root equation

$$cy^2+by+a=0$$

Original roots reciprocalsining equationi.

Shart: c≠0

Shifted roots sum

$$S_k=S+2k$$

xᵢ+k roots sum.

Shifted roots product

$$P_k=P+kS+k^2$$

xᵢ+k roots product.

Scaled roots sum

$$S_k=kS$$

kxᵢ roots sum.

Scaled roots product

$$P_k=k^2P$$

kxᵢ roots product.

Affine root transform

$$y_i=kx_i+h:\quad S_y=kS+2h,\quad P_y=k^2P+khS+h^2$$

Scale va shiftni bitta formula bilan boshqaradi.

Xususiy holatlar: h=0 scale; k=1 shift.

Quadratic conjugate polynomial

$$(x-(p+q\sqrt d))(x-(p-q\sqrt d))=x^2-2px+(p^2-q^2d)$$

Quadratic irrational rootdan rational-coefficient equation tuzadi.

Shart: p,q,d rational, d≥0

Root difference square

$$(x_1-x_2)^2=S^2-4P$$

Orderingga bog‘liq ayirmani square qilib symmetric qiladi.

Xususiy holatlar: |x₁−x₂|=√(S²−4P).

Square-root sum

$$(\sqrt{x_1}+\sqrt{x_2})^2=S+2\sqrt P$$

Positive roots square-root sumini hisoblaydi.

Shart: x₁,x₂≥0

Xususiy holatlar: Principal roots nonnegative.

Reciprocal square-root sum

$$\left(\frac1{\sqrt{x_1}}+\frac1{\sqrt{x_2}}\right)^2=\frac SP+\frac{2}{\sqrt P}$$

Reciprocal radical roots sum.

Shart: x₁,x₂>0

Cubic root sum

$$r_1+r_2+r_3=-\frac ba$$

Cubic first elementary symmetric sum.

Shart: ar³+br²+cr+d=0, a≠0

Cubic pair-product sum

$$r_1r_2+r_2r_3+r_3r_1=\frac ca$$

Cubic second elementary symmetric sum.

Shart: a≠0

Cubic product

$$r_1r_2r_3=-\frac da$$

Cubic total product.

Shart: a≠0

Cubic reciprocal sum

$$\frac1{r_1}+\frac1{r_2}+\frac1{r_3}=\frac{e_2}{e_3}=-\frac cd$$

Uch roots reciprocals sumini koeffitsientlardan beradi.

Shart: d≠0

Cubic square power sum

$$p_2=r_1^2+r_2^2+r_3^2=e_1^2-2e_2$$

Cubic root squares sum.

Cubic cube power sum

$$p_3=e_1^3-3e_1e_2+3e_3$$

Cubic root cubes sum.

Teoremalar va isbotlar

📐 Kvadrat Viyet teoremasi

ax²+bx+c=0 ning ildizlari x₁,x₂ bo‘lsa, x₁+x₂=−b/a va x₁x₂=c/a.

Polynomialni roots factorlari orqali yozib, coefficientlarni taqqoslaymiz.

Isbotni ko'rsatish

Berilgan: ax²+bx+c=0, a≠0, roots x₁,x₂.

Isbotlash kerak: x₁+x₂=−b/a va x₁x₂=c/a.

a(x−x₁)(x−x₂)=0 polynomialini oching.
a[x²−(x₁+x₂)x+x₁x₂]=ax²−a(x₁+x₂)x+a x₁x₂.
Original ax²+bx+c bilan x coefficientlarini tenglashtiring: −a(x₁+x₂)=b.
Constantlarni tenglashtiring: a x₁x₂=c.
a≠0 ga bo‘lib formulalarni oling.

Quadratic Viyet formulalari isbotlandi. ∎

📐 Teskari Viyet teoremasi

Ikki sonning yig‘indisi S va ko‘paytmasi P bo‘lsa, ular x²−Sx+P=0 tenglamaning ildizlaridir.

(x−r₁)(x−r₂) ochilishi aynan x²−Sx+P.

Isbotni ko'rsatish

Berilgan: Ikki son r₁,r₂; S=r₁+r₂, P=r₁r₂.

Isbotlash kerak: r₁,r₂ x²−Sx+P=0 roots ekanini ko‘rsatish.

(x−r₁)(x−r₂)=x²−(r₁+r₂)x+r₁r₂.
r₁+r₂=S va r₁r₂=P ni qo‘ying.
Natija x²−Sx+P.
x=r₁ yoki x=r₂ bo‘lsa productdagi bir factor 0.

Teskari Viyet isbotlandi. ∎

📐 Affine transformed roots teoremasi

x₁,x₂ sum S, product P bo‘lsa, yᵢ=kxᵢ+h roots uchun S_y=kS+2h va P_y=k²P+khS+h².

Yangi rootsni qo‘shish/ko‘paytirish orqali bevosita expand qilinadi.

Isbotni ko'rsatish

Berilgan: x₁+x₂=S, x₁x₂=P va yᵢ=kxᵢ+h.

Isbotlash kerak: S_y=kS+2h va P_y=k²P+khS+h².

y₁+y₂=kx₁+h+kx₂+h=k(x₁+x₂)+2h=kS+2h.
y₁y₂=(kx₁+h)(kx₂+h).
Expand: k²x₁x₂+kh(x₁+x₂)+h².
S,P ni qo‘ying: k²P+khS+h².

Affine transformed-root formulalari isbotlandi. ∎

📐 Quadratic irrational conjugate teoremasi

Ratsional koeffitsientli quadraticda p+q√d (q≠0,d nonsquare rational) root bo‘lsa, ikkinchi root p−q√d bo‘ladi.

Roots sum va product rational bo‘lishi kerak; conjugate irrational termsni bekor qiladi.

Isbotni ko'rsatish

Berilgan: Ratsional koeffitsientli monic quadratic rootsdan biri p+q√d, q≠0.

Isbotlash kerak: Ikkinchi root p−q√d bo‘lishini ko‘rsatish.

Ikkinchi root r bo‘lsin.
Root sum (p+q√d)+r rational bo‘lishi kerak.
q√d irrational qismni bekor qilish uchun r ning irrational qismi −q√d bo‘lishi kerak.
Product ham rational bo‘lishi uchun rational qism p bilan mos conjugate r=p−q√d tanlanadi.
Product p²−q²d rational bo‘ladi.

Quadratic irrational roots conjugate juftlikda keladi. ∎

📐 Cubic Viyet va Newton power-sum extensioni

ar³+br²+cr+d=0 roots r₁,r₂,r₃ uchun e₁=−b/a,e₂=c/a,e₃=−d/a; shundan p₂=e₁²−2e₂ va p₃=e₁³−3e₁e₂+3e₃.

Factor expansion elementary symmetric sumsni, Newton identities power sumsni beradi.

Isbotni ko'rsatish

Berilgan: ar³+br²+cr+d polynomial roots r₁,r₂,r₃.

Isbotlash kerak: Cubic Viyet va p₂,p₃ power-sum formulalarini ko‘rsatish.

Productni ochish x³−e₁x²+e₂x−e₃ beradi.
Coefficient matching: e₁=−b/a, e₂=c/a, e₃=−d/a.
p₂=(r₁+r₂+r₃)²−2(r₁r₂+r₂r₃+r₃r₁)=e₁²−2e₂.
Identity r₁³+r₂³+r₃³=e₁³−3e₁e₂+3e₃ ni expand qilib tekshirish mumkin.
Shuning uchun p₃=e₁³−3e₁e₂+3e₃.

Cubic Viyet va dastlabki Newton power sums isbotlandi. ∎

Yechilgan misollar

oson $-x^2+12x-11=0$ ni monic ko‘rinishga keltiring.

💡 Maslahat: Barcha hadni −1 ga ko‘paytiring.

−1 ga ko‘paytirish tenglama yechimlarini o‘zgartirmaydi.
Natija x²−12x+11=0.

✅ Javob: x^2-12x+11=0

Nega bu usul ishlaydi: Leading coefficient 1 qilinadi va Viyet bevosita o‘qiladi.

⚠️ Faqat x² ishorasini emas, barcha had ishorasini almashtiring.

oson $x^2-5x+4=0$ ildizlari yig‘indisi va ko‘paytmasini toping.

💡 Maslahat: Monic Viyet.

S=−b=5.
P=c=4.

✅ Javob: S=5,\quad P=4

Nega bu usul ishlaydi: x²−Sx+P formasi bilan coefficientlar to‘g‘ridan-to‘g‘ri mos.

⚠️ Middle coefficient −5 bo‘lsa sum +5.

oson $2x^2-7x+4=0$ uchun $S=x_1+x_2$ va $P=x_1x_2$ ni toping.

💡 Maslahat: General Viyetda a ni unutmang.

a=2,b=−7,c=4.
S=−b/a=7/2.
P=c/a=2.

✅ Javob: S=\frac72,\quad P=2

Nega bu usul ishlaydi: Non-monic equationda coefficient ratios ishlatiladi.

⚠️ S=7 deb olish a=2 ni unutishdir.

oson Viyet orqali $x^2-7x+6=0$ ni yeching.

💡 Maslahat: Sum 7, product 6 bo‘lgan ikki sonni toping.

S=7,P=6.
1+6=7 va 1·6=6.

✅ Javob: x=1,6

Nega bu usul ishlaydi: Teskari Viyet integer root pairni tez aniqlaydi.

Muqobil usul: (x−1)(x−6)=0.

⚠️ Sum/product ishoralarini aralashtirmang.

oson Viyet orqali $x^2+6x+8=0$ ni yeching.

💡 Maslahat: Sum −6, product 8.

S=−6,P=8.
−2+(−4)=−6 va (−2)(−4)=8.

✅ Javob: x=-2,-4

Nega bu usul ishlaydi: Ikki manfiy root positive product va negative sum beradi.

⚠️ Rootsni +2,+4 deb olmang.

oson $x^2+2x-1=0$ ildizlari uchun $x_1+x_2+x_1x_2$ ni toping.

💡 Maslahat: Avval S va P.

S=−2, P=−1.
S+P=−2−1.

✅ Javob: -3

Nega bu usul ishlaydi: Expression allaqachon Viyet invariants yig‘indisi.

⚠️ Individual rootsni hisoblash ortiqcha.

oson $x^2+4x-2=0$ ildizlari uchun $1/x_1+1/x_2$ ni toping.

💡 Maslahat: S/P.

S=−4,P=−2.
1/x₁+1/x₂=S/P=(−4)/(−2).

✅ Javob: 2

Nega bu usul ishlaydi: Common denominator x₁x₂ exactly P.

⚠️ P=0 bo‘lsa reciprocal ifoda aniqlanmas edi.

oson $x^2+3x-1=0$ ildizlari uchun $x_1^2x_2+x_2^2x_1$ ni toping.

💡 Maslahat: x₁x₂(x₁+x₂) qilib factorlang.

Expression=P·S.
S=−3,P=−1.
PS=3.

✅ Javob: 3

Nega bu usul ishlaydi: Mixed expression symmetric factorization bilan ikki Viyet invariantiga tushadi.

⚠️ Har bir rootni cube deb noto‘g‘ri o‘qimang.

oson $x^2-3x+1=0$ ildizlari uchun $x_1^3+x_2^3$ ni toping.

💡 Maslahat: S³−3PS.

S=3,P=1.
S³−3PS=27−9.

✅ Javob: 18

Nega bu usul ishlaydi: Cube-sum identity individual radicalsni chetlab o‘tadi.

⚠️ x₁³+x₂³=(x₁+x₂)³ emas; −3PS correction bor.

ortacha $x^2+3x+1=0$ ildizlari uchun $x_1/x_2+x_2/x_1$ ni toping.

💡 Maslahat: (x₁²+x₂²)/P.

S=−3,P=1.
x₁²+x₂²=S²−2P=9−2=7.
Ratio sum=7/P=7.

✅ Javob: 7

Nega bu usul ishlaydi: Ratio expression symmetric common denominatorga keladi.

⚠️ x₁/x₂ alohida symmetric emas, lekin ikki ratio yig‘indisi symmetric.

ortacha $x^2-2x-1=0$ ildizlari uchun $1/(x_1+2)+1/(x_2+2)$ ni toping.

💡 Maslahat: Shifted reciprocal formula, k=2.

S=2,P=−1.
Numerator S+4=6.
Denominator P+2S+4=−1+4+4=7.

✅ Javob: \frac67

Nega bu usul ishlaydi: Denominator (x₁+2)(x₂+2)=P+2S+4.

⚠️ Denominatorni P+4 deb qisqartirib yubormang.

ortacha $x^2+3x-1=0$ ildizlari uchun $1/(x_1+2x_2)+1/(x_2+2x_1)$ ni toping.

💡 Maslahat: Expressionni symmetric common denominatorga keltiring.

S=−3,P=−1.
Numerator 3S=−9.
Denominator 2S²+P=18−1=17.
Natija −9/17.

✅ Javob: -\frac9{17}

Nega bu usul ishlaydi: m=2 formula denominatorni 2S²+P ga soddalashtiradi.

⚠️ Bank 3506 +9/17 deb belgilagan; sign independent algebra bilan −9/17 chiqadi.

ortacha $x^2-4x+1=0$ roots musbat bo‘lsa, $\sqrt{x_1}+\sqrt{x_2}$ ni toping.

💡 Maslahat: Yig‘indini kvadratlang.

S=4,P=1.
(√x₁+√x₂)²=S+2√P=4+2=6.
Principal sum nonnegative.

✅ Javob: \sqrt6

Nega bu usul ishlaydi: Root radicals symmetric square orqali Viyetga tushadi.

⚠️ ±√6 emas; square-root sum nonnegative.

murakkab $x^2-9x+4=0$ roots uchun $1/\sqrt{x_1}+1/\sqrt{x_2}$ ni toping.

💡 Maslahat: Ikkala root musbat; expressionni kvadratlang.

S=9,P=4 va roots musbat.
Square = S/P+2/√P=9/4+1=13/4.
Expression nonnegative.

✅ Javob: \frac{\sqrt{13}}2

Nega bu usul ishlaydi: Reciprocal square-root expression symmetric invariantlarga tushadi.

⚠️ Bank 3512 dagi √2/2 marked answer mos emas; exact QA √13/2 beradi.

ortacha $2x^2+3x-15=0$ roots orasidagi masofani Viyet orqali toping.

💡 Maslahat: (x₁−x₂)²=S²−4P.

S=−3/2,P=−15/2.
S²−4P=9/4+30=129/4.
Masofa = √(129/4).

✅ Javob: \frac{\sqrt{129}}2

Nega bu usul ishlaydi: Root difference square symmetric bo‘lib Viyet bilan topiladi.

⚠️ Masofa absolute; root ordering kerak emas.

oson Ildizlari 2 va 5 bo‘lgan monic kvadrat tenglama tuzing.

💡 Maslahat: S=7,P=10.

Teskari Viyet: x²−Sx+P=0.
S=2+5=7, P=10.

✅ Javob: x^2-7x+10=0

Nega bu usul ishlaydi: Berilgan roots equation factorlari (x−2)(x−5).

⚠️ Middle coefficient −S.

oson Ildizlari −3 va 1 bo‘lgan monic kvadrat tenglama tuzing.

💡 Maslahat: Sum va productni toping.

S=−2,P=−3.
x²−Sx+P=x²+2x−3.

✅ Javob: x^2+2x-3=0

Nega bu usul ishlaydi: Teskari Viyet signsni tizimli boshqaradi.

⚠️ Product −3 ekanini unutmang.

oson $x^2-10x+21=0$ rootsiga teskari sonlar roots bo‘lgan tenglama tuzing.

💡 Maslahat: Coefficientlarni reverse qilish patternini ishlating.

Original a=1,b=−10,c=21.
Reciprocal equation: 21y²−10y+1=0.

✅ Javob: 21y^2-10y+1=0

Nega bu usul ishlaydi: Reciprocal roots sum S/P va product 1/P ga ega.

⚠️ c=0 bo‘lsa reciprocal rootsdan biri aniqlanmaydi.

oson $x^2-15x+50=0$ rootsiga qarama-qarshi sonlar roots bo‘lgan tenglama tuzing.

💡 Maslahat: Sum signi o‘zgaradi, product saqlanadi.

Original S=15,P=50.
New S=−15,P=50.
y²−(−15)y+50=0.

✅ Javob: y^2+15y+50=0

Nega bu usul ishlaydi: Opposite transformda −x substitution odd-degree coefficient signini almashtiradi.

⚠️ Constant signini o‘zgartirmang.

ortacha $x^2-7x+12=0$ rootsidan 5 ga katta roots uchun tenglama tuzing.

💡 Maslahat: Shift formulas.

S=7,P=12,k=5.
S_k=7+10=17.
P_k=12+5·7+25=72.
y²−17y+72=0.

✅ Javob: y^2-17y+72=0

Nega bu usul ishlaydi: Har rootga +5 qo‘shish new sum/productni predictable o‘zgartiradi.

Muqobil usul: Original roots 3,4 → 8,9.

⚠️ Productni P+25 debgina olmang; kS term bor.

ortacha $x^2-8x+15=0$ rootsini 2 marta kattalashtirib yangi tenglama tuzing.

💡 Maslahat: Scale formulas.

S=8,P=15,k=2.
New S=16, new P=60.
y²−16y+60=0.

✅ Javob: y^2-16y+60=0

Nega bu usul ishlaydi: Scale roots sumni k, productni k² marta o‘zgartiradi.

⚠️ Productni faqat 2P=30 deb olmang.

ortacha $5x^2-12x+7=0$ rootsi $x_1,x_2$ bo‘lsa, rootsi $2x_1,2x_2$ bo‘lgan integer-coefficient equation tuzing.

💡 Maslahat: Viyet va scale.

S=12/5,P=7/5.
New S=24/5,P=28/5.
Monic: y²−(24/5)y+28/5=0.
5 ga ko‘paytiring.

✅ Javob: 5y^2-24y+28=0

Nega bu usul ishlaydi: Monic transformed equation nonzero scale bilan integer coefficientsga keltirilishi mumkin.

⚠️ Equationni 5 ga ko‘paytirish rootsni o‘zgartirmaydi.

ortacha Ratsional koeffitsientli monic quadratic rootsidan biri $1+\sqrt2$ bo‘lsa, tenglama tuzing.

💡 Maslahat: Ikkinchi root conjugate.

Ikkinchi root 1−√2.
S=2.
P=1−2=−1.
x²−2x−1=0.

✅ Javob: x^2-2x-1=0

Nega bu usul ishlaydi: Conjugate pair irrational qismlarni sum/productda bekor qiladi.

⚠️ Ikkinchi root −1+√2 emas.

murakkab Ratsional koeffitsientli monic quadratic rootsidan biri $-2+3\sqrt2$ bo‘lsa, tenglama tuzing.

💡 Maslahat: Conjugate −2−3√2.

S=(−2+3√2)+(−2−3√2)=−4.
P=(−2)²−(3√2)²=4−18=−14.
x²−Sx+P=0.

✅ Javob: x^2+4x-14=0

Nega bu usul ishlaydi: Quadratic conjugates rational coefficients hosil qiladi.

⚠️ Productda (a+b)(a−b)=a²−b².

murakkab $x^2-6x+3=0$ roots $x_1,x_2$. Roots $x_1+1/x_2$ va $x_2+1/x_1$ bo‘lgan equation tuzing.

💡 Maslahat: Yangi rootsning sum/productini S=6,P=3 bilan hisoblang.

1/x₁+1/x₂=S/P=2, shuning uchun S_y=S+S/P=8.
P_y=(x₁+1/x₂)(x₂+1/x₁)=P+2+1/P=3+2+1/3=16/3.
y²−8y+16/3=0; 3 ga ko‘paytiring.

✅ Javob: 3y^2-24y+16=0

Nega bu usul ishlaydi: Mixed transform ham symmetric sum/productga tushadi.

⚠️ Cross terms x₁/x₁ va x₂/x₂ ikkalasi ham 1.

murakkab Nechta butun $a$ uchun $x^2+ax+12=0$ ning ikkala ildizi butun son?

💡 Maslahat: Integer roots r,s product 12, a=−(r+s).

Integer factor pairs of 12: (1,12),(2,6),(3,4) va ularning ikkalasi manfiy variantlari.
Positive pairs sums 13,8,7 → a=−13,−8,−7.
Negative pairs sums −13,−8,−7 → a=13,8,7.
Jami 6 distinct a.

✅ Javob: 6

Nega bu usul ishlaydi: Viyet integer-root conditionni divisor enumerationga aylantiradi.

⚠️ Ordered pairsni ikki marta sanamang.

ortacha $2x^3+11x^2+7x+1=0$ roots $r_1,r_2,r_3$. $e_1,e_2,e_3$ ni toping.

💡 Maslahat: Cubic Viyet.

e₁=−b/a=−11/2.
e₂=c/a=7/2.
e₃=−d/a=−1/2.

✅ Javob: e_1=-\frac{11}{2},\ e_2=\frac72,\ e_3=-\frac12

Nega bu usul ishlaydi: Cubic coefficients elementary symmetric sumsni bevosita beradi.

⚠️ Product signi −d/a.

murakkab $x^3+3x^2-5x-2=0$ roots uchun $1/r_1+1/r_2+1/r_3$ ni toping.

💡 Maslahat: e₂/e₃ yoki −c/d.

e₂=−5, e₃=2.
Reciprocal sum=e₂/e₃=−5/2.

✅ Javob: -\frac52

Nega bu usul ishlaydi: Common denominator r₁r₂r₃ numeratorni pair-product sumga aylantiradi.

⚠️ Formula −c/d ham aynan −(−5)/(−2)=−5/2.

murakkab $x^3-5x^2+5x+2=0$ roots kvadratlari yig‘indisini toping.

💡 Maslahat: p₂=e₁²−2e₂.

e₁=5,e₂=5.
p₂=25−10=15.

✅ Javob: 15

Nega bu usul ishlaydi: Three-root square sum ham elementary symmetric quantities orqali hisoblanadi.

⚠️ e₂ pairwise products sum, oddiy product emas.

murakkab $2x^3+6x^2-7x=0$ roots kublari yig‘indisini toping.

💡 Maslahat: Newton/Vieta: e₁=−3,e₂=−7/2,e₃=0.

p₃=e₁³−3e₁e₂+3e₃.
=−27−3·(21/2)+0.
=−27−63/2=−117/2.

✅ Javob: -\frac{117}{2}

Nega bu usul ishlaydi: Power-sum identity rootsni individual yechmasdan ishlaydi.

⚠️ Bank 3574 +117/2 deb belgilagan; Viyet/Newton sign check −117/2 beradi.

Umumiy xatolar

❌ x²+px+q da roots sumini p deb olish.

Viyetda sum −p.

✅ Monic equationni x²−Sx+P shakli bilan solishtiring.

x²−5x+4: S=5.

❌ General ax²+bx+c da a ni unutish.

Non-monic equationda sum/product coefficient ratios.

✅ S=−b/a, P=c/a.

2x²−7x+4: S=7/2, P=2.

❌ Individual rootsni hisoblab keyin symmetric expressionga qo‘yish.

Keraksiz radicals va hisob xatolari paydo bo‘ladi.

✅ Avval expressionni S,P orqali yozing.

x₁³+x₂³=S³−3PS.

❌ x₁³+x₂³=S³ deb yozish.

(x₁+x₂)³ mixed termsni ham o‘z ichiga oladi.

✅ S³−3PS formulasini ishlating.

x²−3x+1 da 18, 27 emas.

❌ 1/x₁+1/x₂=1/S deb olish.

Common denominator P, numerator S.

✅ S/P.

x²+4x−2 da 2.

❌ Ratio sumni S/P deb olish.

x₁/x₂+x₂/x₁ numerator x₁²+x₂².

✅ (S²−2P)/P.

x²+3x+1 da 7.

❌ Shifted roots productini P+k² deb olish.

(x₁+k)(x₂+k) da kS cross-term bor.

✅ P+kS+k².

Roots +5 transformida P_new=P+5S+25.

❌ Scaled roots productini kP deb olish.

Ikkala root ham k ga ko‘payadi.

✅ P_new=k²P.

2x₁,2x₂ product 4P.

❌ Reciprocal-root equationda faqat b ishorasini almashtirish.

Reciprocal transform outer coefficients a va c rollarini almashtiradi.

✅ cx²+bx+a=0.

x²−10x+21 →21x²−10x+1.

❌ Opposite rootsda constant ishorasini ham almashtirish.

Product (−x₁)(−x₂)=P o‘zgarmaydi.

✅ Faqat sum signi almashadi.

x²−15x+50 →x²+15x+50.

❌ Irrational root berilganda ikkinchi rootni ixtiyoriy tanlash.

Ratsional coefficients sum/productda irrational qismni bekor qilishi kerak.

✅ Quadratic conjugate p−q√d ni oling.

1+√2 sherigi 1−√2.

❌ Square-root root expressionsda roots ishorasini tekshirmaslik.

Real principal √xᵢ uchun xᵢ≥0 kerak.

✅ Viyetdan tashqari positivity/domainni tekshiring.

x²−4x+1 roots ikkalasi musbat.

❌ x₁−x₂ ni root ordering berilmasa yagona son deb olish.

Roots nomini almashtirish expression ishorasini o‘zgartiradi.

✅ |x₁−x₂| yoki square kabi symmetric versiyani ishlating, yoxud ordering talab qiling.

x²−6x+1 da √x₁−√x₂ ±2 bo‘lishi mumkin.

❌ Cubicda e₂ ni faqat r₁r₂ deb olish.

e₂ barcha pairwise products yig‘indisi.

✅ r₁r₂+r₂r₃+r₃r₁=c/a.

Cubic Viyetning ikkinchi formulasi.

❌ Cubic product signini d/a deb olish.

Degree 3 da sign alternating: product −d/a.

✅ e₃=−d/a.

x³+4x²+2x−3 da product 3.

❌ Newton power sumda signlarni tekshirmaslik.

p₃=e₁³−3e₁e₂+3e₃.

✅ Elementary symmetric valuesni avval aniq yozing.

2x³+6x²−7x: p₃=−117/2.

❌ Symmetric bo‘lmagan expressionni faqat Viyetdan hisoblash.

Coefficientlar root orderingni belgilamaydi.

✅ Swap test qiling: x₁↔x₂ da qiymat o‘zgarsa qo‘shimcha shart kerak.

3542 kabi unequal-weight pair expression canonical qiymat bermaydi.

❌ Bankdagi marked optionni independent tekshiruvsiz qabul qilish.

Extraction/sign/transcription xatolari mavjud.

✅ S,P identity, expansion yoki substitution bilan marked answerni qayta tekshiring.

3481,3502,3506,3512,3549,3562,3568,3574,3738 kabi savollar flaglangan.

Noto'g'ri tasavvurlar

Viyet faqat integer roots uchun ishlaydi.

Formula real yoki complex roots uchun algebraik multiplicity bilan umumiy ishlaydi.

Viyet bilan rootsning individual qiymatini doim topish mumkin.

Viyet avvalo symmetric sum/productni beradi; individual roots uchun qo‘shimcha yechish kerak bo‘lishi mumkin.

Har root expression S va P bilan aniqlanadi.

Faqat symmetric expressions orderingdan mustaqil ravishda S,P bilan yagona aniqlanadi.

Teskari Viyet faqat oldindan roots ma’lum bo‘lsa ishlaydi.

Sum va product ma’lum bo‘lishi yetarli; rootsni explicit bilish shart emas.

Reciprocal transform har doim mumkin.

Original product P≠0, ya’ni hech bir root 0 emasligi kerak.

Ratsional coefficients bilan bitta irrational root bo‘lsa ikkinchisi ham aynan shu root.

Quadratic irrational odatda conjugate p−q√d bilan juft bo‘ladi.

Cubic Viyet quadratic formulalarning aynan o‘zi.

Cubicda uchta elementary symmetric invariant mavjud: e₁,e₂,e₃.

Newton identities yangi, unrelated teorema.

Ular Viyet bergan elementary symmetric sumsdan power sumsni olishning tizimli davomidir.

Root labels x₁,x₂ canonical tartibga ega.

Agar masalada ildizlar tartibi, masalan x₁ kichik ildiz deb alohida berilmasa, labels almashtirilishi mumkin.

Test bankidagi correct_option matematik isbot o‘rnini bosadi.

Canonical content independent algebraik QA bilan quriladi; marked answer faqat source metadata.

Amaliy qo'llanilishi

Quadratic equation QA

Rootlarni qayta yechmasdan sum/product orqali topilgan javoblarni tez tekshirish mumkin.

Parametrli algebra

Root sum/productga qo‘yilgan shartlar parametr equationlarni diskriminantsiz ham qisqartirishi mumkin.

Polynomial konstruktsiya

Kerakli roots yoki root transforms bo‘yicha yangi polynomial tenglama quriladi.

Contest matematika

Symmetric root expressions va Newton sums olimpiada/sertifikat masalalarini keskin soddalashtiradi.

Signal va control modelling

Characteristic polynomial rootsning sum/producti tizim parametrlarini ifodalashda ishlatiladi.

Sonlar nazariyasi

Integer roots masalalari product divisors va sum constraintsga aylanadi.

Computer algebra

Symbolic systems rootsni explicit radical shaklga ochmasdan symmetric polynomiallar orqali expressionlarni soddalashtiradi.

Test-bank quality assurance

Viyet identities marked optionsdagi sign, coefficient va OCR xatolarini tez aniqlaydi.

Viyet xaritasi: koeffitsientlar ↔ symmetric roots ↔ yangi tenglama

Viyetning asosiy oqimini ko‘rsatadi: coefficientlardan S,P olish, symmetric expressionlarni hisoblash, transformed roots uchun yangi equation tuzish, nonsymmetric ifodalarni swap-test bilan ajratish va cubic/Newton extensionga o‘tish.

Xulosa

Cheat sheet: 1) ax²+bx+c=0: S=x₁+x₂=−b/a, P=x₁x₂=c/a; 2) monic: x²−Sx+P=0; 3) x₁²+x₂²=S²−2P; 4) x₁³+x₂³=S³−3PS; 5) 1/x₁+1/x₂=S/P (P≠0); 6) x₁/x₂+x₂/x₁=(S²−2P)/P; 7) opposite roots: S→−S, P→P; 8) reciprocal roots: S→S/P, P→1/P; 9) shifted roots r+k: S→S+2k, P→P+kS+k²; 10) scaled roots kr: S→kS, P→k²P; 11) cubic ax³+bx²+cx+d: e₁=−b/a, e₂=c/a, e₃=−d/a; 12) symmetric bo‘lmagan ifoda root ordering berilmasa odatda Viyet bilan yagona aniqlanmaydi.

Keyingi “Ratsional tenglamalar” mavzusida denominatorli tenglamalarni domain bilan yechish chuqurlashadi. Keyin yuqori darajali tenglamalar, polynomiallar va parametrli masalalarda Viyet/Newton g‘oyalari yana kengayadi.

Bog'liq mavzular

Oldin bilishingiz kerak: Kvadrat tenglama va uning ildizlari, Qisqa ko'paytirish formulalari, Ko'phadlar va ular ustida amallar

Bog'liq mavzular: Ko'phadlar, Parametrli tenglama va tengsizliklar

Keyingi mavzular: Ratsional tenglamalar, Ikkinchi va yuqori darajali tenglamalar sistemasi, Kvadrat funksiya

Manbalar

Khan Academy — Relation between roots and coefficients — Quadratic root sum/product, proof idea and symmetric expressions.
Khan Academy — Finding the polynomial whose sum and product of roots is given — Inverse Vieta and equation construction from sum/product.
Khan Academy — Finding the unknown through sum and product of roots — Advanced sum/product constraints and parameter-style practice.
OpenStax College Algebra 2e — 2.5 Quadratic Equations — Quadratic standard form, factorization and root/coefficient context.
ProofWiki — Viète’s Formulas — General polynomial Vieta theorem, including explicit cubic coefficient-root formulas.
ProofWiki — Newton’s Identities — Power sums of roots from polynomial coefficients; used for cubic bank-extension.

Shu mavzudagi savollar

$-x^2 + 12x - 11 = 0$ tenglamani normal ko'rinishga keltiring. $3x^2 - 5x + 1 = 0$ tenglamani normal ko'rinishga keltiring. $-2x^2 - 10x + 8 = 0$ tenglamani normal ko'rinishga keltiring. $5x^2 - 10x + 12 = 0$ tenglamani normal ko'rinishga keltiring. $ax^2 + bx + c = 0$ tenglamani normal ko'rinishga keltiring. $x^2 - 5x + 4 = 0$ tenglama ildizlari yig'indisi va ko'paytmasini toping.

Ro'yxatdan o'tib, mashq qilishni boshlang