If | x^n x^n+2 x^n+3 y^n y^n+2 y^n+3 z^n z^n+2 z^n+3 | =(x-y)(y-

step 1 In problem 18 we have the determinant x to the bar of n x the bar of n plus 2 x the bar of n plu

If | x^n x^n+2 x^n+3 y^n y^n+2 y^n+3 ...

If $\left|\begin{array}{ccc}x^{n} & x^{n+2} & x^{n+3} \\ y^{n} & y^{n+2} & y^{n+3} \\ z^{n} & z^{n+2} & z^{n+3}\end{array}\right|$
$=(x-y)(y-z)(z-x),\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right)$ then $n=$
(A) $-2$
(B) $-1$
(C) 0
(D) 1

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Key Concepts

Polynomial Exponents and Factorization

Polynomial exponents and factorization involve manipulating expressions with power functions in algebraic contexts, such as those found in determinants with variable exponents. In problems where unknown exponents are involved, matching the degrees and equating factorizations allows one to solve for the exponents by ensuring the expressions on both sides of an equation are identically equivalent.

Vandermonde Determinant

The Vandermonde determinant is a special form of a determinant that arises when the matrix entries are powers of variables, typically in a sequential order. It has the remarkable property of factoring into a product of differences between the variables. This structure is extensively used in polynomial interpolation and in contexts where the independence of certain variables needs to be demonstrated.

Determinants

Determinants are scalar values computed from square matrices that provide important insights into the properties of the matrix, such as invertibility. They are calculated using specific rules and properties, including linearity and the impact of elementary row operations. Determinants also play a crucial role in solving systems of linear equations and in expressing area or volume scaling factors under linear transformations.

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