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Algebra

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7 Aug 2008 14:57:54 IST

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Polynomial divisibility

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Let g(x) be a non constant polynomial.

Prove that f(x) = xg(x^3) + x-1 is not divisible by x²-x+1

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Rajat Sen

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7 Aug 2008 15:46:34 IST

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Re:Polynomial divisibility

Rajat Sen

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Joined: 7 Aug 2007

Posts: 533

7 Aug 2008 16:36:36 IST

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Is the solution O.K?

Hari Shankar

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7 Aug 2008 19:04:18 IST

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Yet another problem where use of congruences for polynomials makes it an 'inspection problem'.

$x^3-1 = (x+1)(x^2-x+1) \Rightarrow x^3 \equiv 1 \bmod{x^2-x+1}$

Also $x -1 \equiv x - 1 \bmod{x^2-x+1}$

Hence, $x -1 + x g(x^3) \equiv x - 1 + x g(1) \ \bmod{x^2-x+1}$

So, if f(x) is to be divisible by x²-x+1, then x(g(1) +1 ) -1 should be identically zero for all x which is not possible.

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