## Polynomials## Philip J. Erdelsky## March 16, 2009 |

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Polynomials are used in many branches of pure and applied mathematics.

A __polynomial__ in one variable over a field *F* is a
linear combination of a finite number
of nonnegative integral
powers of an independent variable, with coefficients taken from *F*.
The terms are often written in
order of ascending or descending powers:

p(x) = p._{n}x^{ n}+ p_{n-1}x^{ n-1}+ ... + p_{1}x + p_{0}

The order in which the terms are written is irrelevant, because addition
is presumed to be associative and commutative. If a coefficient is zero,
the corresponding term may be omitted. The last term, which is called
the constant term, is actually
*p _{0} x^{ 0}*, but

The polynomial with all coefficients equal to zero is called the
__zero polynomial__ and is exceptional in many ways. A polynomial
with at least one one nonzero coefficient is called a
__nonzero polynomial__.

The polynomial *p(x)* over the field
*F* is also a function
*p(x):F⟶F*,
and if *a* is an element of *F*,

p(a) = p._{n}a^{ n}+ p_{n-1}a^{ n-1}+ ... + p_{1}a + p_{0}

Two polynomials are considered equal if they have equal coefficients
of corresponding powers of the independent variable,
after like terms are combined. If two polynomials are equal in this
sense, then they are equal as functions; i.e., they give equal results
for equal values of the independent variable. The converse is false
for some fields. For example, *x* and
*x ^{ 2}* do not have equal
coefficients, but they are equal for both values in the tiny field

The __degree__ of a nonzero polynomial is the highest exponent with a nonzero
coefficient (after like terms are combined).
For example, *x - 7, x ^{ 3} + x + 1* and

degree | name |
---|---|

0 | constant |

1 | linear |

2 | quadratic |

3 | cubic |

4 | quartic |

5 | quintic |

The zero polynomial has no degree.

The terms "constant" and "linear" are actually a little more general
than this table indicates. The term "constant" includes the zero polynomial,
and the term "linear" includes the zero polynomial and constant polynomials.
Hence a __nonconstant__ polynomial is a nonzero polynomial of degree
at least *1*, and a __nonlinear__ polynomial is a nonzero polynomial
of degree at least *2*.

The set of all polynomials over a field *F* with independent variable
*x* is usually written as *F[x]*.

Addition and multiplication of polynomials are defined to be the usual
elementary algebraic operations. For example, two polynomials are added
by adding the coefficients of corresponding powers of the independent
variable, and two polynomials are multiplied by multiplying all pairs
of terms and combining like terms in the product. When addition and
multiplication are defined in this way, they produce valid results
for all values of the independent variable; i.e.
*(p+q)(x) = p(x) + q(x)*
and *(pq)(x) = p(x)q(x)* for any polynomials over the same field and
any value of *x* in the field.

Multiplication of a polynomial by a constant value from the field can be defined most readily as multiplication of the polynomial by a constant polynomial.

With these definitions, we can show that the polynomials over
a field *F* constitute a vector space over *F*. It
has no dimension, but the set of all polynomials of degree *n*
or less, together with the zero polynomial, constitute a subspace
of dimension *n+1*. The polynomials *1, x, x ^{ 2}, ...,
x^{ n}* are a basis, which is often
called the

The following properties of polynomial addition and multiplication can be derived directly
from the operations on coefficients. Here *p(x)*, *q(x)*
and *r(x)* are polynomials over the same field, and *c*
is an element of the field.

*(p(x)q(x))r(x) = p(x)(q(x)r(x))*(associative law)*(p(x) + q(x)) + r(x) = p(x) + (q(x) + r(x))*(associative law)*p(x)q(x) = q(x)p(x)*(commutative law)*p(x) + q(x) = q(x) + p(x)*(commutative law)*p(x)(q(x)+r(x)) = p(x)q(x) + p(x)r(x)*(distributive law)*c(p(x)q(x)) = (cp(x))q(x)*

In many cases, a nonzero polynomial can be multiplied by any nonzero constant and
still serve the same purpose. It is customary to use the polynomial with
the coefficient of its highest-degree term equal to *1*. Such a
polynomial is called a __monic__ polynomial. The product of two
monic polynomials is monic. If a monic polynomial can be factored
into the product of two or more nonconstant polynomials, then there
is a factorization in which all factors are monic.

The product of two nonzero polynomials is obviously nonzero (and its degree is the sum of the degrees of the factors), a property that the polynomials have in common with the integers. In fact, the polynomials over a field have many properties that are analogous to those of the integers.

The __division algorithm__ for polynomials is as follows.

**Theorem 2.1** *Let p(x) (the *__dividend__*) and d(x)
(the *__divisor__*)
be polynomials over the same
field, where d(x) is nonconstant. Then there are unique polynomials
q(x) (the *__quotient__*) and r(x) (the *__remainder__*) over the same field
such that p(x) = q(x) d(x) + r(x)
and r(x) is zero or of lower degree than d(x).*

*Proof.* We use an induction argument that amounts to long
division. If *p(x)* is zero or the degree of *p(x)* is less than that of *d(x)*,
then
*q(x)* is zero and *r(x)* is *p(x)*.
In other cases, let
*m* be the degree of *d(x)*, let
*n* be the degree of *p(x)*, and write:

d(x) = d_{m}x^{ m}+ t(x),

p(x) = p_{n}x^{ n}+ s(x),

where *t(x)* and *s(x)* are polynomials
of degrees less than *m* and *n*,
respectively (or perhaps zero polynomials).

Then by inductive hypothesis,

s(x) - (p_{n}/d_{m}) x^{ n-m}t(x) = u(x) d(x) + r(x),

where *r(x)* is zero or of degree less than *m*.

Then the required decomposition is

p(x) = ((p_{n}/d_{m}) x^{ n-m}+ u(x)) d(x) + r(x).

Now suppose we have two decompositions:

p(x) = q(x) d(x) + r(x),

p(x) = q'(x) d(x) + r'(x).

Subtract these two equations to obtain:

0 = (q(x) - q'(x)) d(x) + r(x) - r'(x),

(q(x) - q'(x)) d(x) = r'(x) - r(x).

If *q(x) - q'(x)* were nonzero, the degree of the left member would
be greater than the degree of the right member. Hence *q(x) = q'(x)*
and clearly *r(x) = r'(x)*, so the decomposition is unique.
█

The division algorithm could be extended to cases in which the divisor is a nonzero constant polynomial, but it would be rather trivial; the remainder would always be zero.

The polynomial *p(x)* is said to be __divisible__ by
the nonzero polynomial *d(x)*,
or *d(x)* is said to __divide__ *p(x)*,
or *d(x)* is said to be a __divisor__ of *p(x)*,
if
*p(x) = q(x)d(x)* with remainder zero. Many of the properties
of divisibility for integers also apply to polynomials. For example,
if two or more polynomials are divisible by *d(x)*, then their
sum and product are also divisible by *d(x)*.

The following two theorems, called the __remainder theorem__ and
the __factor theorem__, respectively, are immediate consequences of
the division algorithm.

**Theorem 2.2** *If a polynomial p(x) is divided by the linear
polynomial x-a, the remainder is p(a).*

*Proof.* By the division algorithm, *p(x) = q(x)(x-a) + r* for some
constant *r*. Substitution of *a* for *x* gives
*p(a) = r*.
█

**Corollary 2.3** *The polynomial p(x) is divisible by x-a if,
and only if, p(a) = 0.*

The __greatest common divisor__ of two nonzero polynomials is a common
divisor that is divisible by every other common divisor. It is
obviously not unique in most cases, but there is a unique
*monic* greatest common divisor, which is often written
as *gcd(p(x),q(x))*.

The following theorem is the Euclidean algorithm for polynomials. It shows that two nonzero polynomials have a greatest common divisor, and it also exhibits a practical way to compute it.

**Theorem 2.4** *Let p(x) and q(x) be two nonzero polynomials
over the same field. Then they have a monic greatest common divisor g(x),
and there are polynomials s(x) and t(x) such that
*

s(x) p(x) + t(x) q(x) = g(x).

If one polynomial divides the other, the result
is obvious. The divisor is also the greatest common divisor, and
*s(x)* and *t(x)* are
*0* and *1*.

The rest of the proof proceeds by induction on the sum of the
degrees of *p(x)*
and *q(x)*. If the sum is
*0* or *1*, one divides the other.

In other cases, arrange the polynomials so the degree of *p(x)*
does not exceed the degree of *q(x)* and use the divison algorithm
to decompose *q(x)* as follows:

q(x) = u(x) p(x) + r(x),

where *r(x)* is zero or of degree less than the degree
of *p(x)*.

Since *p(x)* does not divide
*q(x)*, *r(x)* must be nonzero.

Any
common divisor of *p(x)* and *q(x)* is also a common divisor of
*p(x)* and *r(x)*, and vice-versa. Hence the greatest common
divisor, if any, is the same in each case. By inductive hypothesis,
*p(x)* and *r(x)* have a monic greatest common divisor *g(x)*,
which is also the monic greatest common divisor of *p(x)* and
*q(x)*, and there are
polynomials *v(x)* and *w(x)* such that

g(x) = v(x) p(x) + w(x) r(x).

But *r(x) = q(x) - u(x) p(x)*, so

g(x) = v(x) p(x) + w(x) (q(x) - u(x) p(x)) = (v(x) - u(x)) p(x) + w(x) q(x),

which is the desired result. █

Two polynomials are __relatively prime__ if their monic greatest
common divisor is *1*.

Modular arithmetic on polynomials is also very similar to modular arithmetic on integers.

Let *M(x)* be a nonzero polynomial. Then two polynomials *p(x)*
and *q(x)*
(over the same field) are __congruent__ with respect to the
__modulus__ *M(x)* if their difference *p(x) - q(x)* is divisible by
*M(x)*. It is easily shown that congruence is an equivalence relation,
and that sums, differences and products of congruent polynomials
(with respect to the same modulus) are congruent.

A polynomial *p(x)* is __irreducible__
or __prime__ if it is nonconstant and
not divisible by any nonconstant polynomial of lower degree.
It is __reducible__ if it can be factored into two (or more)
nonconstant polynomials.

Clearly, all nonconstant linear polynomials are irreducible.

The irreducible polynomials have many properties analogous to those of prime numbers. For example, every nonconstant polynomial is either irreducible or a product of irreducible polynomials.

**Theorem 3.2** *If the product v(x)w(x) of two polynomials is
divisible by an irreducible polynomial p(x), the either v(x) or
w(x) (or both) is divisble by p(x).*

*Proof.* Assume, for purpose of contradiction, that *p(x)*
does not divide either *v(x)* or *w(x)*. Then the monic greatest
common divisor of *p(x)* and *v(x)* is *1*, so by
Theorem 2.4 there are polynomials *s(x)* and *t(x)* such
that

1 = s(x) p(x) + t(x) v(x).

Similarly, there are polynomials *e(x)* and *f(x)* such
that

1 = e(x) p(x) + f(x) w(x).

Multiplication of these two equations gives

1 = s(x) p(x) e(x) p(x) + s(x) p(x) f(x) w(x) + t(x) v(x) e(x) p(x) + t(x) f(x) v(x) w(x).

The left member is not divisible by *p(x)*, but every term
in the right member is divisible by *p(x)*, which is the desired
contradiction.
█

**Corollary 3.3** *If an irreducible polynomial divides the product
of any number of polynomials, it must divide at least one factor.*

The following theorem is analogous to the fundamental theorem of
arithmetic. However, it is *not* called the fundamental theorem
of polynomials!

**Theorem 3.4** *Every monic nonconstant polynomial can be factored into
monic irreducible polynomials in a way that is unique apart
from the order of the factors.*

*Proof.* The proof is by induction on the degree of the polynomial.
A linear polynomial is irreducible, so its factorization is trivial.

In other cases, if the polynomial is irreducible, its factorization is trivial. Otherwise, it can be factored into two monic polynomials of lower degree, each of which has monic irreducible factors by the inductive hypothesis. Therefore, the original polynomial has monic irreducible factors.

To prove uniqueness, assume that a single polynomial has two factorizations into monic irreducible polynomials:

p_{1}(x) p_{2}(x) ... p_{n}(x) = q_{1}(x) q_{2}(x) ... q_{m}(x)

We prove, by induction on the number *n* of polynomials in
the left member, that *m = n* and the right member contains the
same polynomials, perhaps in a different order.

If *n = 1* then

p_{1}(x) = q_{1}(x) q_{2}(x) ... q_{m}(x)

Since *p _{1}(x)* is irreducible,

Now if *n > 1* then *p _{1}(x)* divides the left
member, so it must also divide the right member. By Corollary 3.3

p_{2}(x) p_{3}(x) ... p_{n}(x) = q_{2}(x) q_{3}(x) ... q_{m}(x)

By inductive hypothesis, the two sides contain the same polynomials, perhaps in different orders. This completes the proof. █

If *p(x)* is a polynomial over a field, then a field element *r*
such that *p(r) = 0* is called a __root__ (or __zero__)
of *p(x)*.
Corollary 2.3 states that *r* is a root of *p(x)* if,
and only if, *p(x)* is divisible by *x-r*.

If *p(x)* can be factored as *p(x) = s(x) t(x)*, then the
roots of *p(x)*, if any, are those of the factors *s(x)*
and *t(x)*.

**Theorem 4.1** *A nonzero polynomial of degree n cannot
have more than n roots.*

*Proof.* This is easy to show by induction on *n*. A nonzero
constant polynomial (of degree *0*) obviously has no roots, and
a polynomial of degree *1* obviously has one root.

If *r* is a root of the polynomial *p(x)* of degree *n+1*,
then *p(x) = q(x) (x-r)*, where the degree of *q(x)*
is *n*. Any other root of
*p(x)* must be a root of *q(x)*,
which by inductive hypothesis can have no more than *n*
roots. Hence *p(x)* has at most *n+1* roots.
█

**Corollary 4.2** *A polynomial which is zero or
of degree no greater than n
with more than n roots must be the zero polynomial (when like
terms are combined).*

**Corollary 4.3** *Two polynomials which are zero or of
degree no greater than n
which agree in more than n places must be identical (when like
terms are combined).*

**Corollary 4.4** *Two polynomials over an
infinite field which define the same function must be identical (when like
terms are combined).*

A nonzero polynomial of degree *n* may have fewer
than *n* roots. For example, the polynomial
*(x-1) ^{ 2}* has only one root, although
in this case it is sometimes said that the two roots are equal, or
that the root is a double root.

If *r* is a root of the nonzero polynomial
*p(x)*, then the __degree__
(or __multiplicity__) of the root
is the highest value of *k* such that
*(x-r) ^{ k}* divides

It is easily shown that the sum of the multiplicities of the roots of a nonzero polynomial cannot exceed the degree of the polynomial.

A nonlinear polynomial
may have no roots in the field from which its coefficients
are taken, but it appears that such a polynomial may in some cases
have a root
in some larger field. For example, *x ^{ 2} - 2* has no
roots in the field of rational numbers, but it does have two roots
in the field of real numbers. Similarly,

If *F* is a field, then an __extension__ of
*F* is a field which contains *F*
(or a field isomorphic to *F*) as a subfield.
For example, the complex number field is an extension of the real
number field.

A nonlinear irreducible polynomial can
always be made reducible by extending
the field. This can be done in several (equivalent) ways,
but the simplest is the
process of __adjoining a root__. It is a generalization of the
process by which the field of complex numbers is constructed by
adjoining a root of *x ^{ 2} + 1*.

Let *p(x)* be an irreducible polynomial of degree greater than
*1* with coefficients
in the field *F*. The proposed extension field is the set of
polynomials over *F* of degree less than that of *p(x)*,
together with the zero polynomial.
Addition in this field is defined in the usual way, but multiplication
is taken modulo *p(x)*; i.e., if *s(x)* and *t(x)*
are in the extension field, then their product is the remainder when
*s(x) t(x)* (computed in the usual way) is divided by *p(x)*.
It is fairly easy to show, using the divison algorithm, that addition
and multiplication defined in this way obey all the field axioms
except the existence of a multiplicative inverse for nonconstant
polynomials. To prove that
any nonconstant polynomial *s(x)* has an inverse, we use the Euclidean
algorithm to find polynomials *u(x)* and *v(x)* such that

u(x) s(x) + v(x) p(x) = 1,

u(x) s(x) = (-v(x)) p(x) + 1.

If the degree of *u(x)* is less than that of *p(x)*, it
is the required inverse. Otherwise, use the division algorithm to
find polynomials *q(x)* and *r(x)* such that

u(x) = q(x) p(x) + r(x),

where the degree of *r(x)* is less than that of *p(x)*.
Then a straightforward substitution shows that *r(x)* is the
required inverse.

(q(x) p(x) + r(x)) s(x) = (-v(x)) p(x) + 1,

r(x) s(x) = (-v(x) - q(x)) p(x) + 1.

The constant polynomials constitute a subfield isomorphic to *F*.

Moreover, the linear polynomial *z(x) = x* is a root of *p(x)*
in this field. Hence *p(x)* is divisible by *x - z* in the
extended field, so it is no longer irreducible.

The extension field constructed in this way is also minimal, in the
sense that no proper subfield can contain *F* and
also a root of *p(x)*. Moreover, it is unique up to isomorphism;
i.e., it is isomorphic to any
other minimal extension, and the isomorphism maps each element of
the original field to itself. The required isomorphism is the
mapping which carries the polynomial
*u(x)*
into the value *u(z)*, where
*z* is the constructed root.

If the polynomial is factored in the extended field, it may have nonlinear irreducible factors. We can use the same technique to extend the field further so they, too, are reducible.

The process must stop when an extension field is found in which
*p(x)* can be factored into linear factors. This is called
a __splitting field__ of the polynomial. Every nonconstant
polynomial has at least one. (If the polynomial is not irreducible,
it is factored before beginning the process of adjoining roots.
The splitting field of a linear
polynomial, or of one that can already be factored into linear
factors, is the original field.) The minimal splitting field
of a polynomial is unique, up to isomorphism, and the isomorphism
maps each element of the original field to itself.

The sum of the multiplicities of the roots of a nonconstant polynomial in its splitting field is equal to the degree of the polynomial.

Another feature of field extensions, whether they are obtained by adjoining roots or not, is slightly surprising at first glance. The greatest common divisor of two polynomials is not affected by extending the field, even though each polynomial may acquire additional divisors. However, it is obvious from the Euclidean algorithm that this is so -- the process by which the greatest common divisor is found involves only operations on the original polynomial coefficients, and cannot wander into an extension. In particular, two polynomials that are relatively prime remain so even if the field is extended. The converse, that two polynomials that are not relatively prime remain so if the field is extended, is fairly obvious.

**Theorem 5.1** *Let a be an element of an extension
of the field F which is a root of a nonzero polynomial over F, but
a ∉ F. Then there is
a unique monic irreducible polynomial over F which has a as a root,
and every polynomial over F which has a as a root is divisible by
that unique polynomial.
*

*Proof.* By factoring the polynomial of which
*a* is a root, we can show that *a* is
a root of at least one monic
irreducible polynomial *p(x)* over
*F*.
Assume, for purpose of contradiction,
that it is also a root of a different polynomial *q(x)*
over *F* which is not
divisible by *p(x)*. Since
*p(x)* is
irreducible, the two polynomials are relatively
prime, and by the Euclidean algorithm there are polynomials
*r(x)*
and *s(x)* over *F* such that

Subsitution ofr(x)p(x) + s(x)q(x) = 1.

The uniqueness of *p(x)* is fairly obvious.
Any other monic irreducible polynomial over *F*
with *a* as a root would have to be divisible by
*p(x)*, which is possible only if it is
identical to *p(x)*.
█

We cannot take derivatives of polynomials in the usual sense if the
field is discrete. However, we can define a __formal derivative__
using the formulas from regular differentiation:

p(x) = p_{n}x^{ n}+ p_{n-1}x^{ n-1}+ ... + p_{1}x + p_{0},

p'(x) = n p_{n}x^{ n-1}+ (n-1) p_{n-1}x^{ n-2}+ ... + 2 p_{2}x + p_{1}.

It should be noted that multiplication by the integers *2, 3, ..., n-1, n*
is defined as repeated addition. The integers themselves may not be part of
the field, so regular field multiplication cannot be used.

Formal differentiation has most of the properties of regular differentiation:

- (p(x) + q(x))' = p'(x) + q'(x)
- (c p(x))' = c p'(x)
*(p(x) q(x))' = p'(x) q(x) + p(x) q'(x)*

Some other properties of regular differentiation are not preserved.
For example, polynomials with the same derivative do not necessarily
differ by a constant; e.g. in a field of characteristic *2*,
*x ^{ 2}* and

A nonconstant polynomial is called __separable__ if its roots (in its splitting
field) are all distinct.

A nonconstant linear polynomial is obviously separable. The following results are trivial for linear polynomials.

**Theorem 6.1** *A nonconstant polynomial is separable if, and only if,
it and its derivative are relatively prime.*

*Proof.* No generality is lost by assuming the polynomial is monic.

In the splitting field, we can write the polynomial
as *(x - r _{1})(x - r_{2}) ... (x - r_{n})*,
where

If the polynomial has multiple roots, it can be written as
*(x - r _{1})^{ 2} q(x)*, and its derivative
is

If the roots are distinct, then *(x - r _{1})(x - r_{2})
... (x - r_{n})* is its decomposition into irreducible factors.

Consider any factor *x - r _{i}*. The polynomial can be
written as

**Corollary 6.2** *An irreducible polynomial is separable.*

**Corollary 6.3** *A nonconstant divisor of a separable
polynomial is separable.*

A polynomial in two or more independent variables can be defined as a linear combination of a finite number of products of powers of the independent variables.

The following result for two variables is
the well-known __binomial theorem__.

**Theorem 7.1** *When like terms are gathered together,
the product (x+y) ^{ n} is
*

C(n,0) x^{ n}+ C(n,1) x^{ n-1}y + C(n,2) x^{ n-2}y^{ 2}+ ... + C(n,n-2) x^{ 2}y^{ n-2}+ C(n,n-1) xy^{ n-1}+ C(n,n) y^{ n}.

n (n-1) (n-2) ... (n-k+1) C(n,k) = -------------------------, (1) (2) ... kC(n,0) = C(n,n) = 1, and all the coefficients are integers.

*Proof.*
The proof is by induction on *n*. For *n=1* the result is
obvious.

We then assume the theorem for particular value of *n* and
multiply the expanded form by *x+y* to get the expanded form
of *(x+y) ^{ n+1}*.

The coefficient of *x ^{ n+1-k}y^{ k}* in the product
is

A noteworthy property of the binomial coefficients is that all but the
first and last are
divisible by *n* when *n* is prime.

Another property of rational and real polynomials is useful for quick proofs
of a number of polynomial identities.
Consider two rational or real polynomials which agree for all values
in an interval *[a,b]*, where *a < b*:

p(x) = q(x)fora ≤ x ≤ b.

The interval contains an infinite number of values, so Corollary 4.3
shows that *p(x)* and *q(x)*
must be identical, when like terms are combined (and they must also
be of the same degree).

This can easily be extended to rational or real polynomials of two or more variables by induction on the number of variables.

Suppose

p(xfor_{1},x_{2},...,x_{n}) = q(x_{1},x_{2},...,x_{n})a._{1}≤ x_{1}≤ b_{1}, a_{2}≤ x_{2}≤ b_{2}, ..., a_{n}≤ x_{n}≤ b_{n}

By appropriate algebraic manipulations, each polynomial can be converted into
a polynomial in one variable *x _{n}* with
coefficients as polynomials in the other variables. For example, the
polynomial

Then the result for one variable shows that the each coefficient in one
polynomial is equal to the corresponding coefficient in the other polynomial
for all values of the independent variables in their respective ranges.
By inductive hypothesis, they are identical, when like terms are combined.
This extablishes the result for *n* variables.

Clearly, a similar result holds for complex polynomials, although in this case the variable ranges are two-dimensional.

Now consider the polynomial identity

(p(x) q(x))' = p'(x) q(x) + p(x) q'(x)

The two members are equal for all values of the independent variable
and all values of the coefficients, at least when real polynomials
are involved. Now interpret each side of the identity as a polynomial
in both *x* and the coefficients of
*p(x)*
and *q(x)*. Since the identity holds as an
equation for all values of all the independent variables, the
two sides must be identical when like terms are combined.