Question

Prove that every polynomial of degree n has exactly n-zeros

Answer 1

Answer with explanation:

The The Remainder Theorem states that if a1 is a root of the polynomial equation p(x)=0, then (x−a₁) is a factor of $p_{n}$(x). Using this, we can say that

$p_{n}$(x)=(x−a₁)$p_{n-1}$(x),

where $p_{n-1} (x)$ is a polynomial of degree n−1. We apply the Fundamental Theorem on pn−1(x) to get an expression like

$p_{n} (x)$=(x−a₁)(x−a₂)$p_{n-1} (x)$

Applying this over and over, we get

pn(x)=(x−a₁)(x−a₂)...(x−$a_{n}$)

which proves that a n-th degree polynomial with complex coefficients and will have n roots. However, that does not guarantee that all roots will be unique, though.