Roots of x^3 + ax^2 + bx + c = 0

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SUMMARY

The discussion centers on the polynomial equation \( (x-a)(x-b)(x-c)+1=0 \) and its roots \( \alpha, \beta, \gamma \). Participants debate whether the expression \( (\alpha-x)(\beta-x)(\gamma-x)+1=0 \) holds true for all values of \( x \). A key point of contention arises regarding the validity of the expression when \( x=\alpha \), with one participant questioning its correctness. The conclusion emphasizes the need for clarity in the conditions under which the equation is evaluated.

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If \alpha,\beta\ and\ \gamma are the roots of the eqn.

(x-a)(x-b)(x-c)+1=0

then show that (\alpha-x)(\beta-x)(\gamma-x)+1=0
 
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Do you mean we should show that (\alpha-x)(\beta-x)(\gamma-x)+1=0 for all x? What about x=\alpha, isn't that a counterexample? Maybe I'm missing something vital, but to me it seems incorrect.
 

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