An exercise regarding polynomials

Question

I guess it is a simple exercise though I'm not very good at polynomials. It asks:

Find $m,n,p,q$ natural numbers such that the polynomial $X^m+X^n+X^p+X^q$ is divisible by $x^3+x^2+x+1$. Thank you in advance.

Answer 1

Here is a general method for finding as many such sets of natural numbers as you like:

1. Choose a natural number $k$

2. Multiply $(x^3+x^2+x+1)$ by $x^k$ to get $(x^{k+3}+x^{k+2}+x^{k+1}+x^k)$

3. Obviously, $(x^{k+3}+x^{k+2}+x^{k+1}+x^k)$ is divisible by $(x^3+x^2+x+1)$

4. Then, your natural numbers are $m=k+3$, $n=k+2$, $p=k+1$, $q=k$

Answer 2

Hint: $$1+x+x^2+x^3=(1+x)+x^2(1+x)=(1+x)(1+x^2)$$

Answer 3

Hint : show that $X^m+X^n+X^p+X^q$ is divisible by $X^3+X^2+X+1$ iff $m,n,p,q$ form a complete set of residues modulo $4$. Note that the value of a power $X^j$ modulo $X^3+X^2+X+1$ only depends on the value of $j$ modulo $4$.

In other words, the most general solution to your problem is $X^{4q_1}+X^{4q_2+1}+X^{4q_3+2}+X^{4q_4+3}$ where $q_1,q_2,q_3,q_4$ are nonnegative integers.

Answer 4

A trivial answer is when m=3, n=2, p=1 and q=0 and m=n=p=0 and q=1. This is because the polynomial divides itself.

Non-trivial answers may be obtained by multiplying the polynomial with other polynomial with no constant coefficient, like x, x+x^2, x+x^3, x^2+x^5, etc.

Hope this helps