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Kalkulator Rumus Vieta
Masukkan koefisien polinomial Anda di atas dan klik "Terapkan Rumus Vieta" untuk melihat hasilnya.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! Aturan Vieta berlaku dengan sempurna meskipun akarnya melibatkan bilangan imajiner. Bagian-bagian kompleks hanya menghilangkan satu sama lain selama penjumlahan.
Tidak, ini hanya memberi tahu Anda bagaimana mereka berhubungan satu sama lain sebagai satu kesatuan yang lengkap.
Karena rumus Vieta pada dasarnya mengandalkan normalisasi polinomial (membuat koefisien terdepan 1) terlebih dahulu.
Anda dapat memverifikasi bahwa jumlah akar-akar sama dengan -b/a, jumlah hasil kali berpasangan sama dengan c/a, dan hasil kali semua akar sama dengan -d/a. Ini adalah alat pengecekan kesalahan yang ampuh.
François Viète adalah seorang matematikawan Perancis abad ke-16 yang memelopori penggunaan huruf untuk mewakili hal-hal yang tidak diketahui. Rumusnya yang menghubungkan akar dengan koefisien tetap menjadi landasan aljabar.
