# #| ( (x-1)^4, (x-1)^3, (x-1)^2, (x-1), 1 ), ( (x-2)^4, (x-2)^3, (x-2)^2, (x-2), 1 ), ( (x-3)^4, (x-3)^3, (x-3)^2, (x-3), 1 ), ( (x-4)^4, (x-4)^3, (x-4)^2, (x-4), 1 ), ( (x-5)^4, (x-5)^3, (x-5)^2, (x-5), 1 ) | = # ?

# D = |

( (x-1)^4, (x-1)^3, (x-1)^2, (x-1), 1 ), ( (x-2)^4, (x-2)^3, (x-2)^2, (x-2), 1 ), ( (x-3)^4, (x-3)^3, (x-3)^2, (x-3), 1 ), ( (x-4)^4, (x-4)^3, (x-4)^2, (x-4), 1 ), ( (x-5)^4, (x-5)^3, (x-5)^2, (x-5), 1 ) | #

Write: for Simplicity

The supplied determinant can then be written as follows:

# D = |

( x_1""^4, x_1""^3, x_1""^2, x_1"", 1 ), ( x_2""^4, x_2""^3, x_2""^2, x_2"", 1 ), ( x_3""^4, x_3""^3, x_3""^2, x_3"", 1 ), ( x_4""^4, x_4""^3, x_4""^2, x_4"", 1 ), ( x_5""^4, x_5""^3, x_5""^2, x_5"", 1 ) | #

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The given expression appears to represent a 5x5 matrix with entries derived from polynomial expressions involving ( x ). Each row consists of powers of ( x ) subtracted by constants from 1 to 5, and each column represents different powers of these expressions, from ( (x-a)^4 ) to ( 1 ), where ( a ) ranges from 1 to 5.

To simplify the matrix, we can expand each polynomial expression:

For the first row: [ (x-1)^4 = x^4 - 4x^3 + 6x^2 - 4x + 1 ] [ (x-1)^3 = x^3 - 3x^2 + 3x - 1 ] [ (x-1)^2 = x^2 - 2x + 1 ] [ (x-1) = x - 1 ] [ 1 = 1 ]

We apply the same expansion process to the remaining rows, replacing ( x ) with ( x-a ) for each row from 2 to 5.

For the second row: [ (x-2)^4 = x^4 - 8x^3 + 24x^2 - 32x + 16 ] [ (x-2)^3 = x^3 - 6x^2 + 12x - 8 ] [ (x-2)^2 = x^2 - 4x + 4 ] [ (x-2) = x - 2 ] [ 1 = 1 ]

For the third row: [ (x-3)^4 = x^4 - 12x^3 + 54x^2 - 108x + 81 ] [ (x-3)^3 = x^3 - 9x^2 + 27x - 27 ] [ (x-3)^2 = x^2 - 6x + 9 ] [ (x-3) = x - 3 ] [ 1 = 1 ]

For the fourth row: [ (x-4)^4 = x^4 - 16x^3 + 96x^2 - 256x + 256 ] [ (x-4)^3 = x^3 - 12x^2 + 48x - 64 ] [ (x-4)^2 = x^2 - 8x + 16 ] [ (x-4) = x - 4 ] [ 1 = 1 ]

For the fifth row: [ (x-5)^4 = x^4 - 20x^3 + 150x^2 - 500x + 625 ] [ (x-5)^3 = x^3 - 15x^2 + 75x - 125 ] [ (x-5)^2 = x^2 - 10x + 25 ] [ (x-5) = x - 5 ] [ 1 = 1 ]

Thus, the expanded matrix is:

[ \begin{pmatrix} x^4 - 4x^3 + 6x^2 - 4x + 1 & x^3 - 3x^2 + 3x - 1 & x^2 - 2x + 1 & x - 1 & 1 \ x^4 - 8x^3 + 24x^2 - 32x + 16 & x^3 - 6x^2 + 12x - 8 & x^2 - 4x + 4 & x - 2 & 1 \ x^4 - 12x^3 + 54x^2 - 108x + 81 & x^3 - 9x^2 + 27x - 27 & x^2 - 6x + 9 & x - 3 & 1 \ x^4 - 16x^3 + 96x^2 - 256x + 256 & x^3 - 12x^2 + 48x - 64 & x^2 - 8x + 16 & x - 4 & 1 \ x^4 - 20x^3 + 150x^2 - 500x + 625 & x^3 - 15x^2 + 75x - 125 & x^2 - 10x + 25 & x - 5 & 1 \ \end{pmatrix} ]

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