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Kalkulator Formula Vieta
Masukkan pekali polinomial anda di atas dan klik "Gunakan Formula Vieta" untuk melihat keputusan.Vieta's formulas establish elegant relationships between the roots of a polynomial and its coefficients, without requiring you to solve the equation first. For a cubic ax³ + bx² + cx + d = 0 with roots r&sub1;, r&sub2;, r&sub3;, the formulas state: the sum of roots r&sub1;+r&sub2;+r&sub3; = −b/a, the sum of pairwise products r&sub1;r&sub2;+r&sub1;r&sub3;+r&sub2;r&sub3; = c/a, and the product of all roots r&sub1;r&sub2;r&sub3; = −d/a.
These formulas are named after François Viète (1540–1603), a French mathematician who pioneered using letters for unknowns. The formulas arise naturally from expanding the factored form a(x−r&sub1;)(x−r&sub2;)(x−r&sub3;) and comparing coefficients with the standard form. They work identically whether the roots are real or complex.
Vieta's formulas serve two critical purposes: error checking (verify that your computed roots are consistent with the original coefficients) and indirect computation (compute symmetric functions of the roots without knowing the roots individually). They are foundational in competition mathematics, abstract algebra, and numerical analysis.
After solving a cubic, check that the sum and product of your roots match −b/a and −d/a respectively.
Many olympiad problems ask about symmetric functions of roots without requiring you to find the roots explicitly.
Vieta's formulas help detect numerical instability — if computed roots don't satisfy the formulas, precision is lost.
The sum of roots is NEGATIVE b/a, and the product is NEGATIVE d/a. Missing these minus signs is extremely common.
All formulas require dividing by the leading coefficient a. If a ≠ 1, the raw coefficient is NOT the answer.
Vieta's formulas work identically for complex roots. The sum and product relationships hold universally.
| Sum of Roots | r&sub1;+r&sub2;+r&sub3; = −b/a |
| Pairwise Products | r&sub1;r&sub2;+r&sub1;r&sub3;+r&sub2;r&sub3; = c/a |
| Product of Roots | r&sub1;·r&sub2;·r&sub3; = −d/a |
| Named After | François Viète (1540–1603) |
| Works With | Both real and complex roots |
Ya! Peraturan Vieta terpakai dengan sempurna walaupun akarnya melibatkan nombor khayalan. Bahagian kompleks hanya membatalkan satu sama lain semasa penambahan.
Tidak, ia hanya memberitahu anda bagaimana ia berkaitan antara satu sama lain sebagai satu set lengkap.
Kerana formula Vieta sememangnya bergantung pada menormalkan polinomial (menjadikan pekali utama 1) terlebih dahulu.
Anda boleh mengesahkan bahawa jumlah punca sama dengan -b/a, jumlah hasil berpasangan sama dengan c/a, dan hasil darab semua punca bersamaan -d/a. Ia adalah alat semakan ralat yang berkuasa.
François Viète ialah seorang ahli matematik Perancis abad ke-16 yang mempelopori penggunaan huruf untuk mewakili perkara yang tidak diketahui. Formula beliau yang menghubungkan punca kepada pekali kekal sebagai asas algebra.
