Rearranging the conditional constraint gives x – 2y – 6 = 0. The polynomial can be formatted as x cube + (-2y) cube + (-6) cube – 3(x)(-2y)(-6), which evaluates directly to 0 because the base sum is 0.
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We first find a square + b square + c square by expanding (a + b + c) square, which equals 5. Substituting these evaluated numerical components into the standard three-variable cubic identity gives minus 25.
: Setting up the equation as x + 1/x = 10/3 creates a quadratic form 3x square – 10x + 3 = 0. Factoring this gives (3x – 1)(x – 3) = 0, which provides the two possible numerical answers.
We first take out the common constant factor 6 from the volume expression. This leaves a square – 4b square inside brackets, which breaks down further into (a + 2b)(a – 2b) using the difference of squares formula.
This six-term polynomial matches the three-term square identity. Looking at the negative signs, the bases for both v and w must be negative to yield the correct cross-product signs.