I'm having a bit trouble with this excercise:
The problem:
Let there be a polynomial $f(x)=a_1x^{t_1} + a_2x^{t_2} + ... + a_nx^{t_n}$ Where $t_1, t_2, ..., t_n$ are not-negative integers. The polynomial has a root $b$ which occurs $n$ times. Prove that $b = 0$.
What I have so far:
I can presume that $a_1, a_2, ..., a_n \neq 0$
If $n = 1$, then it's obviously true.
If $n = 2$, I tried using this:
$f^{(n-1)}(b) = 0 \\ f^{(n)}(b) = 0$
So:
$f(b) = a_1b^{t_1} + a_2b^{t_2} = 0 \\ f'(b) = t_1*a_1b^{(t_1-1)} + t_2*a_2b^{(t_2-1)} = 0 \\ f''(b) = (t_1-1)t_1*a_1b^{(t_1-2)} + (t_2-1)t_2*a_2b^{(t_2-2)} \neq 0$
Why can't $b \neq 0$ be true?