Complex numbers
Lab: equations, factors and multiplicities
Solve a quadratic, build a polynomial from roots or test a candidate root. Check each factor with a step-by-step division.
Learning-path index · Read the article · Stage 6: all complex roots
Problem and result
Complex plane
Automatic zoom on nearby roots. Tick labels show actual coordinates: the origin may lie outside the window. Points are not artificially displaced.
One point per distinct root; m indicates multiplicity, not coordinate size. Orange = selected. In verification, a red cross marks a candidate that is not a root.
Roots and multiplicities
≈ marks approximate plot coordinates; the formula and discriminant remain exact.
How the product expands
Division and Horner scheme
One coefficient per second. Choose a step with the slider or table; on the plot use left/right arrows, Home/End and Enter on points.
All steps (scroll horizontally on mobile)
| j | Previous result | Product by r | Coefficient to add | New result |
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