WEBVTT

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Here is the problem: 2x to the power 4 minus 3x cubed plus 5x squared minus 6x minus 5 equals 0.

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A polynomial equation of degree 4 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 4 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 0.5172794, x equals 1.6472505; complex: 0.18501447 plus or minus 1.7028612i.

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A degree-4 polynomial has exactly 4 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 0.5172794: left side equals 0, right side equals 0, which checks out.

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x equals 1.6472505: left side equals 0, right side equals 0, which checks out.