Website owner: James Miller
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 ai’s are in the field F is called a polynomial in λ over the field F.
Zero polynomial. If every ai = 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
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].
Ayres. Matrices (Schaum).
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Jesus Christ and His Teachings
Words of Wisdom
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On integrity and the lack of it
The test of a person's Christianity is what he is
Who will go to heaven?
The superior person
On faith and works
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
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On modern intellectualism
On Self-sufficient Country Living, Homesteading
Principles for Living Life
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
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We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
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Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Walking a solitary road
Intellectual disparities among people and the power in good habits
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On responding to wrongs
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Sizing up people
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Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life
We are our habits
What creates moral character?