Polynomial challenge: Show that not all the coefficients of f(x) are integers.

In summary, $f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$. It is shown that not all the coefficients of $f(x)$ are integers. Assuming all coefficients to be integers, it is found that $r-p\mid p-q$ instead of $r-q\mid p-q$. Congratulations! :)
  • #1
castor28
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$f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$.

Show that not all the coefficients of $f(x)$ are integers.
 
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  • #2
castor28 said:
$f(x)$ is a degree 10 polynomial such that $f(p)=q$, $f(q)=r$, $f(r)=p$, where $p$, $q$, $r$ are integers with $p<q<r$.

Show that not all the coefficients of $f(x)$ are integers.

Let all coefficients be integers
we have m-n divides $f(m)-f(n)$
so p- q | f(p) - f(q) | q-r

similarly
q -r | r-p
and r-p | p - q

from above as p - q | q-r | r- p | p- q (all nteger multiples) so all are same and hence a contradiction

so all coefficients cannot be integers
 
Last edited:
  • #3
kaliprasad said:
Let all coefficients be integers
we have m-n divides $f(m)-f(n)$
so p- q | f(p) - f(q) | q-r

similarly
q -r | r-p
and r-q | p - q

from above as p - q | q-r | r- p | p- q (all nteger multiples) so all are same and hence a contradiction

so all coefficients cannot be integers
That is quite correct (except for a small typo: $r-q\mid p-q$ should be $r-p\mid p-q$).
Congratulations!:)
 
  • #4
castor28 said:
That is quite correct (except for a small typo: $r-q\mid p-q$ should be $r-p\mid p-q$).
Congratulations!:)

Thanks castor. Corrected the same inline for the flow.
 

Related to Polynomial challenge: Show that not all the coefficients of f(x) are integers.

1. What is a polynomial?

A polynomial is an algebraic expression consisting of variables and coefficients, typically combined using addition, subtraction, and multiplication. For example, f(x) = 3x^2 + 2x + 1 is a polynomial with variables x and coefficients 3, 2, and 1.

2. What does it mean for a polynomial to have integer coefficients?

A polynomial with integer coefficients means that all the numbers in the expression (including variables raised to a power) are whole numbers, either positive, negative, or zero.

3. How can you show that not all coefficients of a polynomial are integers?

One way to show this is by providing a counterexample. For instance, you can find a polynomial with at least one coefficient that is not an integer, such as f(x) = 2.5x^2 + 1. This disproves the idea that all coefficients must be integers.

4. Why is it important to prove that not all coefficients of a polynomial are integers?

Proving that not all coefficients of a polynomial are integers can help us better understand the properties and limitations of polynomials. It also allows us to explore different types of numbers and their relationships, such as rational and irrational numbers.

5. Can a polynomial have only non-integer coefficients?

Yes, a polynomial can have only non-integer coefficients. For example, f(x) = √2x^2 + πx + e is a polynomial with non-integer coefficients. This type of polynomial is called an irrational polynomial.

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