A polynomial of degree n has at most n zeros

Dependencies:

  1. Field
  2. Factor theorem
  3. Product of linear factors is a factor
  4. Degree of factor is less than degree of polynomial

Let $F$ be a field. Let $p(x) \in F[x]$ and $\deg(p) = n$. Then $p$ has at most $n$ zeros.

Proof

Let $a_1, a_2, \ldots, a_k$ be zeros of $p$. By factor theorem, $(x-a_i) \mid p(x)$ for all $1 \le i \le k$. This means $r(x) = \prod_{i=1}^k (x - a_i)$ divides $p(x)$.

Since $r$ is a factor of $p$, $\deg(r) \le \deg(p) \implies k \le n$.

Dependency for: None

Info:

Transitive dependencies:

  1. Group
  2. Ring
  3. Polynomial
  4. Degree of product of polynomials
  5. Zero divisors of a polynomial
  6. Degree of factor is less than degree of polynomial
  7. Polynomial divisibility
  8. Degree of sum of polynomials
  9. Integral Domain
  10. 0x = 0 = x0
  11. Field
  12. A field is an integral domain
  13. Polynomial division theorem
  14. Factor theorem
  15. Product of linear factors is a factor