Polynomial with integer coefficients

In summary, a polynomial with integer coefficients is a mathematical expression consisting of variables, coefficients, and exponents where all coefficients are integers. The main difference between polynomials with integer coefficients and those with non-integer coefficients is that the coefficients in the former are all integers while the latter may include fractions and decimals. Examples of polynomials with integer coefficients include 4x^3 + 2x^2 + 5x + 1, -3x^2 + 7x - 2, and x^4 + 10x^2 + 9. To simplify a polynomial with integer coefficients, you can combine like terms and use the distributive property. In science, polynomials with integer coefficients are used in various applications
  • #1
anemone
Gold Member
MHB
POTW Director
3,883
115
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.
 
Mathematics news on Phys.org
  • #2
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
 

FAQ: Polynomial with integer coefficients

What is a polynomial with integer coefficients?

A polynomial with integer coefficients is a mathematical expression that consists of variables, coefficients, and operations such as addition, subtraction, and multiplication. The coefficients in this type of polynomial are all integers, which means they are whole numbers without any fractional or decimal parts.

How do you identify a polynomial with integer coefficients?

To identify a polynomial with integer coefficients, you need to check the coefficients of each term. If all the coefficients are integers, then the polynomial has integer coefficients. For example, 3x^2 + 5x + 2 is a polynomial with integer coefficients because all the coefficients (3, 5, and 2) are integers.

What is the degree of a polynomial with integer coefficients?

The degree of a polynomial with integer coefficients is the highest exponent of the variable in the polynomial. For example, the degree of 3x^2 + 5x + 2 is 2 because the highest exponent of x is 2.

Can a polynomial with integer coefficients have negative exponents?

No, a polynomial with integer coefficients cannot have negative exponents. This is because the exponents in a polynomial represent the number of times the variable is multiplied by itself, and having a negative exponent would result in a fraction, which is not allowed in a polynomial with integer coefficients.

How are polynomials with integer coefficients used in science?

Polynomials with integer coefficients are used in various scientific fields, including physics, chemistry, and engineering. They are used to model and describe real-world phenomena, such as the motion of objects, chemical reactions, and electrical circuits. They are also used in data analysis and optimization problems in scientific research.

Similar threads

B Finding polynomials with given roots
Replies
14
Views
1K
MHB Find Polynomials Fulfilling Real Coefficient Equation
Replies
1
Views
975
MHB Polynomial challenge: Show that not all the coefficients of f(x) are integers.
Replies
3
Views
1K
MHB Expressing a polynomial P(x)=(x−a)^2(x−b)^2+1 by two other polynomials
Replies
2
Views
1K
MHB Proof: polynomial with integer solutions
Replies
8
Views
3K
I Primes and Polynomials
Replies
26
Views
3K
A Prime Number Powers of Integers and Fermat's Last Theorem
Replies
1
Views
1K
MHB Prove No Integers Solve $ax^3+bx^2+cx+d=1$ for x=19,2 for x=62
Replies
3
Views
1K
MHB Find Polynomials: Real Coeffs Resulting in 1
Replies
1
Views
854
MHB First, second and third derivatives of a polynomial
Replies
1
Views
1K

Hot Threads

Recent Insights

Back
Top