#1203 · Algebra · Polynomials

Symmetric Product of Linear Forms

If $a x + cy + b z = X$ , $c x + b y + a z = Y$ , and $b x + a y + cz = Z$ , show that $(a^{2} + b^{2} + c^{2} - b c - c a - ab) (x^{2} + y^{2} + z^{2} - y z - z x - x y) = X^{2} + Y^{2} + Z^{2} - Y Z - Z X - X Y$ .

~20 minProof · checked by AIalgebraic identitycube roots of unityfactorization

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. VIII. Surds and Imaginary Quantities, Examples VIII.b, No. 31

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