x³ − 6x² + 11x − 6
- Coefficients, highest power first
- 1, -6, 11, -6
- Roots
- 1, 2, 3
- Real roots (counted with multiplicity)
- 3
- Largest real root
- 3
- Smallest real root
- 1
مصدر التحقق: (x − 1)(x − 2)(x − 3) expanded by hand
حاسبة جذور كثيرات الحدود
All real and complex roots of a polynomial up to degree 6, such as a cubic or quartic equation, with repeated roots found exactly and 30-digit accuracy.
آخر تحديث أمثلة تم التحقق منها: 7
Roots
1, 2, 3
Roots: 1, 2, 3
- Real roots (counted with multiplicity)
- 3
- Largest real root
- 3
- Smallest real root
- 1
- Degree
- 3
This degree-3 polynomial has 3 real roots (counting multiplicity).
p(x) = x³ − 6x² + 11x − 6
Roots in the complex plane (dashed: unit circle)
Roots
| # | Real part | Imaginary part | Multiplicity | |p(root)| |
|---|---|---|---|---|
| 1 | 1 | 0 | 1 | 0 |
| 2 | 2 | 0 | 1 | 0 |
| 3 | 3 | 0 | 1 | 0 |
طريقة الحساب S
Polynomial
Durand–Kerner iteration
Started from powers of 0.4 + 0.9i scaled by the Cauchy bound; 12 sweeps until every correction was below 10⁻³⁰ of the root, then Newton polishing.
Exact rational roots
Each was confirmed by substituting the fraction into p(x) and getting exactly 0.
Check with Vieta's formulas
حول حاسبة جذور كثيرات الحدود
A polynomial of degree n has exactly n roots among the complex numbers, counted with multiplicity; this is the fundamental theorem of algebra. The calculator first splits off repeated factors exactly with Yun's square-free algorithm, then finds all remaining roots at once with the Durand–Kerner (Weierstrass) iteration, which refines n guesses together until each correction is below 10⁻³⁰ of the root. Rational roots are confirmed by exact substitution.
Cubic and quartic equations from engineering, physics and algebra courses are typical inputs. The default, 1, −6, 11, −6, is x³ − 6x² + 11x − 6 = (x − 1)(x − 2)(x − 3), so the roots are 1, 2 and 3; they sum to 6 and multiply to 6, as Vieta's formulas require.
Enter coefficients from the highest power down, with zeros for missing powers: x³ − 2 is 1, 0, 0, −2. Degrees 1 to 6 are accepted.
أمثلة محلولة
x³ − 2 (one real, two complex)
- Coefficients, highest power first
- 1, 0, 0, -2
- Roots
- 1.25992105, −0.6299605249 ± 1.091123636i
- Real roots (counted with multiplicity)
- 1
- Largest real root
- 1.2599210499
مصدر التحقق: Python decimal: 2^(1/3) and 2^(1/3)·(−1/2 ± i√3/2)
x⁴ − 1
- Coefficients, highest power first
- 1, 0, 0, 0, -1
- Roots
- −1, 1, ±i
- Real roots (counted with multiplicity)
- 2
مصدر التحقق: Fourth roots of unity: ±1, ±i
Repeated root (x − 1)³(x + 2)
- Coefficients, highest power first
- 1, -1, -3, 5, -2
- Roots
- −2, 1 (×3)
- Real roots (counted with multiplicity)
- 4
- Smallest real root
- -2
مصدر التحقق: (x − 1)³(x + 2) = x⁴ − x³ − 3x² + 5x − 2 (expanded with Python fractions)
الأسئلة
How many roots does a polynomial have?
Exactly as many as its degree, counted with multiplicity, once complex roots are included; this is the fundamental theorem of algebra. x⁴ − 1 has four roots: −1, 1, i and −i. The number of real roots can be smaller: x³ − 2 has one real root, ∛2 ≈ 1.259921, and two complex ones. A root of multiplicity 3, such as x = 1 in (x − 1)³(x + 2), counts three times.
How do you solve a cubic equation?
Look for a rational root first. By the rational root theorem, any rational root p/q of a polynomial with integer coefficients has p dividing the constant term and q dividing the leading coefficient. For x³ − 6x² + 11x − 6, trying divisors of 6 finds x = 1, and dividing by (x − 1) leaves x² − 5x + 6 = (x − 2)(x − 3). Without a rational root, Cardano's formula or a numerical method is needed.
Is there a formula for the roots of a quintic?
No general formula using radicals exists for degree 5 or higher. The Abel–Ruffini theorem, proved by Abel in 1824, shows this, and Galois theory explains which equations can be solved that way. x⁵ − x − 1 is a standard example whose roots cannot be written with radicals. Numerical methods still find them: its only real root is about 1.167304.
What are Vieta's formulas?
They link the roots to the coefficients. For aₙxⁿ + aₙ₋₁xⁿ⁻¹ + … + a₀, the roots add up to −aₙ₋₁/aₙ and multiply to (−1)ⁿa₀/aₙ. For x³ − 6x² + 11x − 6 the roots 1, 2 and 3 sum to 6 and multiply to 6, matching −(−6)/1 and (−1)³ × (−6)/1. The calculator uses both as a check on the roots it finds.
Why do complex roots come in conjugate pairs?
When every coefficient is real, conjugating the equation p(z) = 0 gives p(z̄) = 0, so the conjugate of a root is also a root. x³ − 2 therefore has the pair −0.629961 ± 1.091124i alongside its real root. It follows that a polynomial of odd degree with real coefficients always has at least one real root.
ما مدى دقة «حاسبة جذور كثيرات الحدود»؟
تعتمد الدقة على مدخلاتك وافتراضات الطريقة. يستخدم الحساب العشري 50 رقمًا معنويًا، لكن التقديرات والأساليب العددية وبيانات المصدر قد تكون أقل دقة؛ تقريب القيم المعروضة لا يزيل هذه الحدود. أمثلة محلولة جرى التحقق منها بمصادر مستقلة: 7. مثلًا، يجري التحقق من «x³ − 6x² + 11x − 6» بالرجوع إلى (x − 1)(x − 2)(x − 3) expanded by hand.
ما مصدر هذه الطريقة؟
Wolfram MathWorld — Durand-Kerner Method (Weierstrass iteration); D. Y. Y. Yun, On square-free decomposition algorithms, SYMSAC 1976.
حول هذه الحاسبة
المصادر
تم التحقق بالرجوع إلى المصادر
تتضمن هذه الحاسبة أمثلة محلولة بإجابات من مصادر مستقلة، وعددها 7. تُشغّل ضمن مجموعة الاختبارات، ويمكنك تشغيلها هنا أيضًا.
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