Using properties of determinants show that:
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c1- c1 + c2 + c3
| ( 1 + x+ x^2). x. x^2
| ( 1 + x+ x^2). 1. x
|( 1 + x + x^2). x^2 . 1
(1 + x + x^2) | 1 x x^2
| 1. 1. x
| 1. x^2. 1
( 1 + x+ x^2)(( 1 - x^3 - x( 1 - x) + x^2( x^2 -1))
( 1 + x + x^2)( ( 1 - x^3 - x+ x^2 + x^4 - x^2))
( 1 +x + x^2)(( 1 - x^3 - x+ x^4))
1 - x^3 - x+ x^4 + x- X^4 - x^2 + x^5 + x^2 - x^5 - x^3 + x^6
1 - 2x^3 + x^6
( x^3)^2 + 1 - 2x^3
( 1 - x^3)^2
| ( 1 + x+ x^2). x. x^2
| ( 1 + x+ x^2). 1. x
|( 1 + x + x^2). x^2 . 1
(1 + x + x^2) | 1 x x^2
| 1. 1. x
| 1. x^2. 1
( 1 + x+ x^2)(( 1 - x^3 - x( 1 - x) + x^2( x^2 -1))
( 1 + x + x^2)( ( 1 - x^3 - x+ x^2 + x^4 - x^2))
( 1 +x + x^2)(( 1 - x^3 - x+ x^4))
1 - x^3 - x+ x^4 + x- X^4 - x^2 + x^5 + x^2 - x^5 - x^3 + x^6
1 - 2x^3 + x^6
( x^3)^2 + 1 - 2x^3
( 1 - x^3)^2
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