Proving $f(x)=0$ Has No Integer Solution with Integer Coefficients

In summary, a polynomial $f(x)$ with integer coefficients and odd values for $f(0)$ and $f(1)$ has no integer solutions for $f(x) = 0$. This is because the constant term must be odd and there must be an even number of odd coefficients in the non-constant terms, making the overall result odd for any integer input. Therefore, no integer value can satisfy the equation.
  • #1
juantheron
247
1
A polynomial $f(x)$ has Integer Coefficients such that $f(0)$ and $f(1)$ are both odd numbers. prove that $f(x) = 0$ has no Integer solution
 
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  • #2
jacks said:
A polynomial $f(x)$ has Integer Coefficients such that $f(0)$ and $f(1)$ are both odd numbers. prove that $f(x) = 0$ has no Integer solution

There are details you may need to fill in yourself but:

\(f(0)\) odd implies that the constant term is odd

then \(f(1)\) odd implies that there are an even number of odd coeficients of the non constant terms.

So if \(x \in \mathbb{Z}\) then \(f(x)\) is odd and so cannot be a root of \(f(x)\) CB
 

1. How do you prove that a function has no integer solution?

To prove that a function has no integer solution, you must first analyze the function and its coefficients. Then, you can use various methods such as the Rational Root Theorem, substitution, and contradiction to show that there are no possible integer values that satisfy the equation.

2. What are some common techniques used to prove that $f(x)=0$ has no integer solution?

Some common techniques used to prove that $f(x)=0$ has no integer solution include the Rational Root Theorem, substitution, and contradiction. Other methods such as the Remainder Theorem, Euclidean Algorithm, and modular arithmetic can also be useful.

3. Can a function have no integer solution but still have real solutions?

Yes, a function can have no integer solution but still have real solutions. For example, the function $f(x)=x^2+1$ has no integer solutions but has two real solutions, $x=1$ and $x=-1$.

4. Are there any special cases where a function with integer coefficients has no integer solution?

Yes, there are special cases where a function with integer coefficients has no integer solution. One example is when the function has no real solutions, such as $f(x)=x^2+1$, as mentioned earlier. Another example is when the function has complex solutions, such as $f(x)=x^2+4$, which has two complex solutions, $x=2i$ and $x=-2i$.

5. Is there a guaranteed method to prove that a function has no integer solution?

No, there is no guaranteed method to prove that a function has no integer solution. However, by utilizing various techniques and methods, it is possible to prove that a function has no integer solution with a high degree of confidence. It is important to carefully analyze the function and its coefficients and use multiple approaches to support the proof.

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