’s are in the field F is called a polynomial in
over the field F.Polynomials over a field, Polynomial domain, Quotients of polynomials, Remainder theorem, Greatest common divisor, Unique Factorization Theorem
Polynomial in λ over a field F. Let λ denote an abstract symbol (indeterminate) which is assumed to be commutative with itself and with the elements of a field F. The expression
(1) f(λ) = anλn + an-1λn-1 + ... + a1λ + a0λ0
where the
’s are in the field F is called a polynomial in
over the field F.
Zero polynomial. If every
= 0 in (1) above it is called the zero polynomial and is
written f(λ) = 0.
Leading term of a polynomial. The term of highest degree i.e. the term anλn in (1) above..
Monic polynomial. A polynomial of the form
f(λ) = λn + an-1λn-1 + ... + a1λ + a0λ0
where the coefficient of the leading term is 1.
Equal polynomials. Two polynomials which contain the same terms.
Polynomial domain F[λ] over a field F. The set of all polynomials in λ (of every degree: n = 0, 1, 2, ... , ∞) over the field F – the totality of all such polynomials.
The individual polynomials within the polynomial domain F[λ] can be added, subtracted and multiplied and the domain can be viewed as an abstract number system. Regarded as such it meets all the axiomatic requirements of a field except that it lacks a multiplicative identity element. It meets the axiomatic requirements of an integral domain. For example, the following laws hold:
f(λ) + g(λ) = g(λ) + f(λ)
( commutative over addition )
f(λ) ∙ g(λ) = g(λ) ∙ f(λ)
( commutative over multiplication )
If f(λ)
0 while f(λ) ∙ g(λ) = 0 , then g(λ) = 0
( no divisors of zero )
If g(λ)
0 and h(λ)∙ g(λ) = k(λ)∙ g(λ) , then h(λ) = k(λ)
( cancellation law holds )
Quotients of polynomials.
Theorem (Division algorithm). Let f(λ) and g(λ)
0 be polynomials in the polynomial
domain F[λ] over the field F. Then there exist unique polynomials h(λ) and r(λ) in F[λ], where
r(λ) is either the zero polynomial or is of degree less than that of g(λ), such that
(2) f(λ) = h(λ) ∙ g(λ) + r(λ)
Here r(λ) is called the remainder in the division of f(λ) by g(λ), If r(λ) = 0 , g(λ) is said to divide f(λ) and g(λ) and h(λ) are called factors of f(λ).
Note that the above theorem is exactly analogous to the division algorithm for integers: For any integer a and any positive integer b, there exist unique integers q and r such that
a = bq + r, 0
r < b
The integer a is the dividend, b is the divisor, q is the quotient and r is the remainder. [This theorem could be stated differently as “the quotient a/b equals q plus a remainder of r” – which explains the terminology.]
Let f(λ) = h(λ) ∙ g(λ). When g(λ) is of degree zero, that is, when g(λ) = c, a constant, the factorization is called trivial. A non-constant polynomial over F is called irreducible over F if its only factorization is trivial.
Irreducible polynomial. A polynomial that cannot be written as the product of two polynomials with degrees of at least 1 and having coefficients in some given domain or field. Unless otherwise stated, “irreducible” means irreducible in the field of the coefficients of the polynomial under consideration.
Remainder theorem. When a polynomial in x is divided by x - h, the remainder is equal to the number obtained by substituting h for x in the polynomial.. More concisely, f(x) = (x-h)q(x) + f(h), where q(x) is the quotient and f(h) the remainder.
Example. If f(x) = x4 - 2x3 + 3x2 - x + 2 is divided by x - 3, the quotient is q(x) = x3 + x2 + 6x + 17 and the remainder is 53. If we substitute h = 3 into f(x) we get 34 - 2 ∙ 33 + 3 ∙ 32 - 3 + 2 = 53.
Proof. Let a polynomial f(x) be divided by x - h until a constant term is obtained. Then (2) above becomes
(3) f(x) = (x - h)g(x) + r
where g(x) is the quotient and r is a constant. If we substitute x = h into (3) we get
f(h) = (h - h)g(x) + r = 0 ∙ g(x) + r = r
Common divisor of two or more polynomials. A polynomial which is a factor of each of the polynomials. If h(x) divides both f(x) and g(x), it is called a common divisor of f(x) and g(x).
Greatest common divisor of two or more polynomials. A polynomial d(x) is called the greatest common divisor of polynomials f1(x), f2(x), ... ,fn(x) if all the following hold:
● d(x) is monic
● d(x) is a common divisor of all the polynomials f1(x), f2(x), ... ,fn(x)
● every common divisor of f1(x), f2(x), ... ,fn(x) is a divisor of d(x)
Theorem 1. If f(x) and g(x) are polynomials in F[x], not both the zero polynomial, they have a unique greatest common divisor d(x) and there exist polynomials h(x) and k(x) in F[x] such that
d(x) = h(x) ∙ f(x) + k(x) ∙ g(x) .
This theorem has a counterpart in the realm of the integers: Any two integers a
0 and b
0
have a positive greatest common divisor (g.c.d.) which can be expressed as a “linear
combination” of a and b in the form d = au + bv for integers u and v.
When the only common divisors of f(x) and g(x) are constants, their greatest common divisor is d(x) = 1.
Theorem 2. If the greatest common divisor of f(x) of degree n > 0 and g(x) of degree m > 0 is not 1, there exist non-zero polynomials a(x) of degree < m and b(x) of degree < n such that
a(x) ∙ f(x) + b(x) ∙ g(x) = 0
and conversely.
Relatively prime polynomials. Two polynomials are called relatively prime if their greatest common divisor is 1.
Theorem 3. If g(x) is irreducible in F[x] and f(x) is any polynomial of F[x], then either g(x) divides f(x) or g(x) is relatively prime to f(x).
Theorem 4. If g(x) is irreducible but divides f(x) ∙ h(x), it divides at least one of f(x) and h(x).
Theorem 5. If f(x) and g(x) are relatively prime and if each divides h(x), so also does f(x) ∙ g(x).
Unique Factorization Theorem. Every non-zero polynomial f(x) of F[x] can be written as
f(x) = c ∙ q1(x)∙ q2(x)∙ ... ∙ qr(x)
where c
0 is a constant and the qi(x) are monic irreducible polynomials of F[x].
References.
Ayres. Matrices (Schaum).