If the number of roots of a polynomial are more than degree of polynomial, then it becomes an identity. Why?

Question

This is more like only well-known fact to me, but no one really coherently explained it to me. I’ve only done questions for proving that a polynomial is an identity: for example, in the case of a quadratic, all the three standard coefficients should be zero for it to be an identity. I wanted more intuition for this. Can someone explain the whole topic and relevant surrounding stuff to me?

Answer 1

We can always write a polynomial in terms of its roots. Suppose it is of degree $n$ and has leading coefficient $a$: $$a(x-r_1)(x-r_2)\cdots(x-r_n)$$ That there are $n$ factors is a result of the fundamental theorem of algebra and the factor theorem.

From this we see that if $a\ne0$, there can only be the roots $r_1,r_2,\dots,r_n$ – everywhere else, all the factors and $a$ are non-zero, which multiply into a non-zero product. Contraposition then gives the desired result: if there are more roots, $a=0$ and the polynomial collapses into the identity.

Answer 2

A very important theorem, you have to know is the foundamental theorem of algebra:

A polynomial $P(x)$ of degree $n$ with complex or real coefficents have no more than $n$ zeros each counted with its multiplicity.

Now, it's very easy to see that a polynomial written in the form: $$a_nx^n+\cdots+a_0=0 \rightarrow a(x-x_1)\cdots(x-x_n)$$ can't have more than $n$ real or complex solutions.