Ruffini's Theorem

Theorem 5.4.1. (Ruffini's Theorem) Let f(x) ∈ K[x] and α ∈ K. α is a zero of f i.e. f(α) = 0 if and only if (x − α) | f(x).

Proof. There is a polynomial q such that

If α is a root of f(x) then evaluating eq. (5.4.1) for x = α we have f(α) = 0 = (α − α)q(x) + r, thus r must be equal to zero, so f(x) is divisible by (x − α).

Conversely, if f is divisible by (x − α), then we have

then obviously f(α) = 0.□

Definition 5.4.2. Let f(x) ∈ 𝕂[x]. An element α ∈ 𝕂[x] such that f(α) = 0 is said root of f(x).

Definition 5.4.3. A root α ∈ K of f(x) ∈ 𝕂[x] is said to be simple if (x − α) | f(x) but (x − α)² ∤ f(x). It is said to be of multiplicity m if (x − α)^m | f(x), but (x − α)^{m + 1} ∤ f(x). A root of multiplicity m > 1 is said to be a multiple root.

Theorem 5.4.4. Let K a field. A nonzero polynomial f(x) of degree n in 𝕂[x], has at most n roots ∈ 𝕂, counted with their multiplicity.

Proof. We proceed by induction on n. If n = 0, f(x) is a non-zero constant without roots, hence the theorem is true. We suppose true the theorem for any polynomial of degree < n. Let f(x) of degree n. If f(x) has a root α of multiplicity m ≥ 1, m ≤ n, then

If β is another root of f(x) then 0 = f(β) = (β − α)^m q(β). Since β α we must have q(β) = 0, because 𝕂 has no zero-divisors. Thus the only roots of f(x) are α (with multiplicity m) and the roots of q(x). For the induction hypothesis, q(x) has at last n − m roots (= degree ∂q(x)). Hence f(x) can have at last m + (n − m) = n roots in 𝕂. □

Corollary 5.4.5. Let 𝕂 a field with infinitely many elements and f(x) and g(x) two polynomials ∈ 𝕂[x]. Then they are equal as polynomials f(x) = g(x) only if they are equal as polynomial functions f(x) = g(x) on 𝕂.

Proof. If f(x) = g(x) as polynomials, then they will obviously equal as polynomial function. Conversely, suppose that f(α) = g(α), ∀α ∈ 𝕂. Then the polynomial h(x) = f(x) − g(x) is the polynomial with the property that h(α) = 0, ∀α ∈ 𝕂. If h(x) wasn't the zero polynomial, it would have degree n ∈ ℕ. Since 𝕂 has infinite elements, h(x) would be a polynomial with infinite roots. Then h(α) must be the zero-polynomial and f(x) = g(x) as polynomials.□

With the last we have proved that the notions of polynomial functions and polynomials are the same.

Exercise 5.4.6. Let us determine whether each of the following polynomials is irreducible over ℤ₅.

1. f(x) = x³ + 2x² − 3x + 4

2. g(x) = x² + 3x + 4

Routine computations show that

Thus 3 is a zero of f(x) in ℤ₅, and f(x) is reducible over ℤ₅.

In the case of, g(x) is irreducible over ℤ₅ since g(x) has no zeros in ℤ₅: