WEBVTT

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Here is the problem: x to the power 6 minus 64 equals 0.

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A polynomial equation of degree 6 in x.

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Start with the equation.

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Move every term to one side.

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Test candidates from the rational root theorem: x equals negative 2 is a root. Synthetic division by (x plus 2) leaves a polynomial of degree 5.

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Test candidates from the rational root theorem: x equals 2 is a root. Synthetic division by (x minus 2) leaves a polynomial of degree 4.

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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 2, x equals 2; complex: 1 plus or minus 1.7320508i, negative 1 plus or minus 1.7320508i.

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A degree-6 polynomial has exactly 6 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 2: left side equals 0, right side equals 0, which checks out.

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x equals 2: left side equals 0, right side equals 0, which checks out.