Find every root of a polynomial from its coefficients, highest power first — for x³ − 6x² + 11x − 6 you would enter 1, −6, 11, −6.
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From the article Polynomial Roots — Quadratic, Cubic and Quartic Explained.
Example 1 — 2x² − 7x + 3 = 0
a = 2, b = −7, c = 3.
D = (−7)² − 4(2)(3) = 49 − 24 = 25, and √25 = 5.
x = (7 ± 5) / 4 → x = 12/4 = 3 or x = 2/4 =
0.5
Check by substitution: 2(9) − 7(3) + 3 = 18 − 21 + 3 = 0 ✓ and 2(0.25) − 7(0.5) + 3 = 0.5 − 3.5 + 3 = 0 ✓
Example 2 — complex roots: x² + 2x + 5 = 0
D = 4 − 20 = −16, so √D = 4i.
x = (−2 ± 4i) / 2 = −1 + 2i or
−1 − 2i — a conjugate pair, as expected.
Check: (−1 + 2i)² = 1 − 4i + 4i² = 1 − 4i − 4 = −3 − 4i. Then (−3 − 4i) + 2(−1 + 2i) + 5 = −3 − 4i − 2 + 4i + 5 = 0 ✓
Worked in full in Polynomial Roots — Quadratic, Cubic and Quartic Explained.
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