Polynomial with integer coefficients

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SUMMARY

In the discussion, it is established that for a polynomial \( P \) with integer coefficients and three distinct integers \( a, b, c \), the simultaneous conditions \( P(a)=b, P(b)=c, P(c)=a \) cannot be satisfied. This conclusion is derived from the properties of polynomials and integer mappings, demonstrating that such a configuration leads to contradictions in the values of \( P \). The proof relies on the fundamental theorem of algebra and the nature of polynomial functions over integers.

PREREQUISITES
  • Understanding of polynomial functions and their properties
  • Familiarity with integer coefficients in polynomials
  • Knowledge of the fundamental theorem of algebra
  • Basic concepts of mappings and function behavior
NEXT STEPS
  • Explore the implications of the fundamental theorem of algebra on polynomial equations
  • Research integer polynomial mappings and their characteristics
  • Investigate examples of polynomial functions with integer coefficients
  • Learn about contradictions in mathematical proofs and their significance
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Mathematicians, students studying algebra, and anyone interested in the properties of polynomials with integer coefficients.

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Let $a,\,b,\,c$ be three distinct integers and $P$ be a polynomial with integer coefficients. Show that in this case the conditions $P(a)=b,\,P(b)=c,\,P(c)=a$ cannot be satisfied simultaneously.
 
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The polynomial is P(x)

we have m-n divides P(m) - p(n)

Let the given condition is true

So $a-b | P(a) - p(b)$

or $a-b | b- c$

Similarly

$b - c | c- a$

And $ c- a | a - b$

From above 3 have

$a-b | b-c | c- a | a-b$So all are same and hence a = b= c which is a contradiction.so condition can not be satisfied simultaneously
 

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