WEBVTT

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Here is the problem: x cubed minus 3x plus 1 equals 0.

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A polynomial equation of degree 3 in x.

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Start with the equation.

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Move every term to one side.

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No further rational roots; approximate the remaining 3 roots numerically. The Durand–Kerner method iterates on all roots at once until they stop changing.

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Collect the real solutions.

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Now look at the graph of the left side minus the right side.

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It crosses zero exactly at the solutions we found.

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So the answer is x equals negative 1.8793852, x equals 0.34729636, x equals 1.5320889.

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A degree-3 polynomial has exactly 3 roots counted with multiplicity (some may be complex). Exact rational roots are found first, then the rest are solved exactly or approximated.

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x equals negative 1.8793852: left side equals 0, right side equals 0, which checks out.

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x equals 0.34729636: left side equals 0, right side equals 0, which checks out.

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x equals 1.5320889: left side equals 0, right side equals 0, which checks out.