Factor theorem

Dependencies:

1. Polynomial division theorem

Let $F$ be a field. Let $p(x) \in F[x]$ and $a \in F$.

$p(a) = 0 \iff (x-a) \mid p(x)$

Proof of 'only if' part

Let $p(a) = 0$.

By the division theorem, there is a unique $(q, r)$ such that $p(x) = (x-a)q(x) + r(x)$ such that $\deg(r) < \deg(x-a) = 1$.

$\deg(r) < 1 \Rightarrow r \in F$. Therefore, $p(x) = (x-a)q(x) + r$.

$ 0 = p(a) = (a-a)q(a) + r = r$.

Therefore, $p(x) = (x-a)q(x) \Rightarrow (x-a) \mid \mid p(x)$.

Proof of 'if' part

$(x-a) \mid p(x) \Rightarrow p(x) = (x-a)q(x)$ for some $q(x) \in F[x]$.

Therefore, $p(a) = (a-a)q(a) = 0$.

Dependency for:

1. Product of linear factors is a factor
2. A polynomial of degree n has at most n zeros

Info:

• Depth: 6
• Number of transitive dependencies: 11

Transitive dependencies:

1. Group
2. Ring
3. Polynomial
4. Degree of product of polynomials
5. Zero divisors of a polynomial
6. Degree of sum of polynomials
7. Integral Domain
8. 0x = 0 = x0
9. Field
10. A field is an integral domain
11. Polynomial division theorem