Find the number of monic polynomials $p(x)$ with complex coefficients such that degree of $p(x)$ is at most $7$ and all its roots lie in the set {${1,2,3}$}.
I know the definition of monic polynomial here
I've done :
when degree of monic polynomial is $1$ we have $p(x)= x+a$ , in this case we can get $3$ monic polynomials because roots lie in {${1,2,3}$}.
when degree is $2$ , $p(x)=x^2+ax+b$ , in this case we have ? monic polynomials
But I am unable to think further .
Could anyone please help me ?? Any short method would be highly appreciated .(as Im preparing for an exam where we have to answer within $3$ minutes)
Thanks!
(I'm not sure about the tags .)
All those polynomials have the form $$p(x)=(x-1)^m(x-2)^n(x-3)^p$$
That's because when working over the complex number you will always have that the polynomial will split into linear factors.
If you impose the condition that $\deg(p)\leq7$, then this translates into $m+n+p\leq7$
Thus you need to solve $m+n+p\leq7$ in the non-negative integers.
Can you do this?