[Question 2014a01] solution 3 (upgrade method, provided by Dong Qilin)

[Question 2014a01] solution 3 (upgrade method, provided by Dong Qilin)

Introduce the variable \ (Y \) and upgrade \ (| A | \). Consider the following factors:

\ [| B | = \ begin {vmatrix} 1 & x_1-a & x_1 (x_1-a) & X_1 ^ 2 (x_1-a) & \ cdots & X_1 ^ {n-1} (x_1-a) \ 1 & x_2-a & X_2 (x_2-a) & X_2 ^ 2 (x_2-a) & \ cdots & X_2 ^ {n-1} (x_2-a) \ vdots & \ vdots \ 1 & x_n-a & X_n (x_n-a) & x_n ^ 2 (x_n-a) & \ cdots & x_n ^ {n-1} (x_n-a) \ 1 & Y-A (Y-a) & y ^ 2 (Y-) & \ cdots & y ^ {n-1} (Y-a) \ end {vmatrix }. \]

Add \ (| B | \) to the next column by multiplying \ (A \) in sequence.

\ [| B | =\begin {vmatrix} 1 & X_1 & X_1 ^ 2 & X_1 ^ 3 & \ cdots & X_1 ^ n \ 1 & X_2 & X_2 ^ 2 & X_2 ^ 3 & \ cdots & X_2 ^ n \\\ vdots & \ vdots \ 1 & x_n ^ 2 & x_n ^ 3 & \ cdots & x_n ^ n \ 1 & Y & y ^ 2 & y ^ 3 & \ cdots & y ^ n \ end {vmatrix} = \ prod _ {1 \ leq I <J \ Leq n} (x_j-x_ I) \ prod _ {I = 1} ^ N (y-x_ I ). \ cdots (1) \]

On the other hand, expand \ (| B | \) according to the last row.

\ [| B | = (-1) ^ n \ prod _ {1 \ Leq I <J \ Leq n} (x_j-x_ I) \ prod _ {I = 1} ^ N (x_i-a) + (-1) ^ {n + 1} | A | (Y-A) + Y (Y-) d, \ cdots (2) \]

The expansion of the \ (n-1 \) item after the last line presents the common factor \ (Y-a) \), and the remaining part is recorded as \ (d \) (It is not important ). consider (1) and (2) as polynomials about \ (Y \). When \ (A \ NEQ 0 \), compare its constant term (in other words, order \ (y = 0 \),

\ [\ Prod _ {1 \ Leq I <J \ Leq n} (x_j-x_ I) \ prod _ {I = 1} ^ nx_ I = \ prod _ {1 \ Leq I <J \ Leq n} (x_j-x_ I) \ prod _ {I = 1} ^ N (x_i-a) + A |, \]

Therefore

\ [| A | =\ frac {1} {A} \ prod _ {1 \ Leq I <J \ Leq n} (x_j-x_ I) \ big (\ prod _ {I = 1} ^ nx_ I-\ prod _ {I = 1} ^ N (x_i-a) \ big ). \]

When \ (a = 0 \), compare the coefficient before the item \ (Y \), there are

\ [| A | = \ prod _ {1 \ Leq I <J \ Leq n} (x_j-x_ I) \ big (\ sum _ {I = 1} ^ nx_1 \ cdots \ hat {x} _ I \ cdots x_n \ big ). \ quad \ Box \]

[Question 2014a01] solution 3 (upgrade method, provided by Dong Qilin)