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

Click For Summary
A polynomial \(f(x)\) with integer coefficients has \(f(0)\) and \(f(1)\) as odd numbers, indicating that the constant term is odd and there is an even number of odd coefficients among the non-constant terms. This structure ensures that for any integer \(x\), \(f(x)\) remains odd. Consequently, since odd numbers cannot equal zero, \(f(x) = 0\) cannot have any integer solutions. Therefore, it is proven that \(f(x) = 0\) has no integer solutions.
juantheron
Messages
243
Reaction score
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
 
Mathematics news on Phys.org
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
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
View full post »

Similar threads

MHB Polynomial with integer coefficients
  • · Replies 1 ·
Replies
1
Views
2K
MHB Proof: polynomial with integer solutions
  • · Replies 8 ·
Replies
8
Views
3K
MHB Polynomial challenge: Show that not all the coefficients of f(x) are integers.
  • · Replies 3 ·
Replies
3
Views
1K
MHB Compute $\sum_{n=1}^{2020} (f(n)-g(n))$
  • · Replies 1 ·
Replies
1
Views
1K
Graduate Prime Number Powers of Integers and Fermat's Last Theorem
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Proof of a fact about real numbers - transcendental and algebraic
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Understanding Algebraic Numbers & Proving Them
  • · Replies 1 ·
Replies
1
Views
2K
MHB Find Polynomials Fulfilling Real Coefficient Equation
  • · Replies 1 ·
Replies
1
Views
1K
MHB Positive Integer Solutions of $(x^2+y^2)^n=(xy)^{2014}$
  • · Replies 2 ·
Replies
2
Views
2K
MHB Prove that the sum of 6 positive integers is a composite number
  • · Replies 1 ·
Replies
1
Views
1K