**§ ****5 ****Polynomials**

[ Integer-valued polynomial ] When the variable *x* is an integer , the value of a polynomial *f* ( *x* ) is always an integer, and this polynomial is called an integer-valued polynomial.

Integer-coefficient polynomials are a special case of integer-valued polynomials.

Integer-valued polynomial expressions:

1 ° where the *nth* integer polynomial can be expressed as

_{}

where is an integer,_{}

_{}

2 ° integer-valued odd polynomial ( satisfying *f* ( - *x* ) *=* - *f* ( *x* )) must be expressed as

_{}

where is an integer ._{}

A 3 ° integer-valued even polynomial ( satisfying *f* ( - *x* ) *= f* ( *x* )) must be expressed as

_{}_{}

where is an integer ._{}

[ Reducible and irreducible polynomials ] Let *f* ( *x* ) be a polynomial with rational coefficients , if there are non-constant polynomials with rational coefficients *g* ( *x* ) and *h* ( *x* ), such that

*f* ( *x* ) *= g* ( *x* ) *h* ( *x* )

*Then f* ( *x* ) is called reducible ( or reducible) in the field of rational numbers , otherwise *f* ( *x* ) is called an irreducible polynomial over the field of rational numbers ( abbreviated as irreducible polynomial ).

[ Gauss's theorem ] Let *f* ( *x* ) be an integer coefficient polynomial , which is reducible in the rational number field , then there must be two integer coefficient polynomials *g* ( *x* ) and *h* ( *x* ), such that

*f* ( *x* ) *= g* ( *x* ) *h* ( *x* )

[ Eisenstein's test ]

1 ° is set as . If there is a prime number *p* , such that _{}

_{}but _{}

Then *f* ( *x* ) is an irreducible polynomial .

2 ° is set as a polynomial of degree *2n+* 1 integer coefficient , if there is a prime number *p* , such that _{}

_{}

but _{}

Then *f* ( *x* ) is an irreducible polynomial .

[ Pylon's Discrimination Method ]

1 ° set

_{}

is an integer coefficient polynomial of degree *n* with a coefficient of 1 , which satisfies the conditions :

(i)
_{}

(ii)
_{}

(iii) _{}Real numbers )

Then *f* ( *x* ) is an irreducible polynomial .

2 ° set

_{}

is an integer coefficient polynomial of degree *n* with a coefficient of 1 , which satisfies the conditions :

(i)
_{}

(ii) _{}

Then *f* ( *x* ) is an irreducible polynomial .

3 ° set

_{}

is an integer coefficient polynomial of degree *n* with a coefficient of 1 , which satisfies the conditions :

_{}

Then *f* ( *x* ) is an irreducible polynomial .

4 ° set

_{}

*n* with a coefficient of 1 , which satisfies the conditions :

_{}

_{}

Then *f* ( *x* ) is an irreducible polynomial .

5 ° set

_{}

is an integer coefficient polynomial of degree *n* whose first term is 1 and whose constant term is not zero , and satisfies the conditions :

_{}

Then *f* ( *x* ) is an irreducible polynomial .

[ Divisibility of Polynomials ] Let *f* ( *x* ) and *g* ( *x* ) be polynomials with two rational coefficients , and *g* ( *x* ) is not always zero , if there is a polynomial *h* ( *x* ) such that

*f* ( *x* ) *= g* ( *x* ) *h* ( *x* )

*Then g* ( *x* ) is said to be divisible by *f* ( *x* ), denoted as

_{}or _{}

At this time , *g* ( *x* ) is called a factor of *f* ( *x* ) , and *f* ( *x* ) is called a multiple of *g* ( *x* ) . Otherwise , *g* ( *x* ) cannot divide *f* ( *x* ), which is written as ._{}

The following ¶ ° *f* represents the degree of the polynomial *f* ( *x* ) .

The divisibility of polynomials has the following properties :

1 ° _{}

2 ° If and , then *f* and *g* differ only by a constant factor . _{}_{}

3 ° If , then _{}_{}_{}

4 ° if , then¶ ° *f *¶ ° *g *_ _{}_{}

If , and , then *f* is called a true factor of *g* , obviously ¶ ° *f < *¶ ° *g.*_{}_{}

5 ° If *p* ( *x* ) is an irreducible polynomial , and , then or . _{}_{}_{}

6 ° If *p* ( *x* ) is an irreducible polynomial , and

*f* ( *x* ) *=* 0, *p* ( *x* ) *=* 0

If there is a common root , there must be ._{}

[ Polynomial with remainder division ] Let *f* ( *x* ) *and g* ( *x* ) be arbitrary polynomials , and *g* ( *x* ) is not always zero , then there must be two polynomials *q* ( *x* ) and *r* ( *x* ) *such* that

*f* ( *x* ) *= g* ( *x* ) *q* ( *x* ) *+r* ( *x* )

where *r* ( *x* ) *=* 0 or ¶ ° *r< *¶ ° *g.* This is called polynomial division with remainder .

[ Polynomial rolling division method ] The definition of polynomial rolling and rolling division method and integer rolling division method are completely similar , and it is only necessary to regard the literal symbols in the formula in § 1(1) of this chapter as polynomials .

Similarly , the unique decomposition theorem of polynomials , the highest common factor and the lowest common multiple, and the concepts and formulas of polynomial coprime are completely similar to those in the section on integers , and it is only necessary to regard the symbols in the corresponding formulas as polynomials .

Example Polynomial

_{} and_{}

The highest common factor of .

Solution To avoid fractions , multiply *f* ( *x* ) by 2 , then divide f ( x *) **by **g* ( *x* ) :

_{}

_{}
_{}

During the calculation , the first difference is multiplied by 2 , so the quotient is changed , but the remainder only obtains a number factor of 2, which does not matter . Multiply *g* ( *x* ) by 3 , and divide by :_{}_{}

_{}

_{}_{}

desirable_{}

_{}

So the required common factor is ._{}

[ Congruence ]

1 ° polynomial mode congruence Let *m* ( *x* ) be a polynomial , if

_{}

*Then f* ( *x* ) and *g* ( *x* ) are said to be congruent modulo *m* ( *x* ) , denoted as

_{}

2 ° Prime Modulo Congruence Let *p* be a prime number , *f* ( *x* ) and *g* ( *x* ) are polynomials with integer coefficients , if the corresponding coefficients are all congruent modulo *p* , then the two polynomials are said to be congruent modulo p *,* and write do

_{}

3 ° Multiple modular congruence formula Let *p* be a prime number , ( *x* ) is a polynomial , if *f* ( *x* ) - *g* ( *x* ) is a multiple of ( *x* ) , mod p *,* then *f* ( *x* ) and *g* ( *x* ) Congruence modulo *p,* ( *x* ) , denoted as _{}_{}* *_{}

*f* ( *x* ) *g* ( *x* )_{}
_{}

[ Promotion of Fermat's Theorem ] Let *p* be a prime number , ( *x* ) * *_{}is an irreducible polynomial of degree *n* , mod *p,* then for any polynomial that is not a multiple of ( *x* ) *f* ( *x* ), mod *p,* there is always_{}

_{}_{}1_{}

For any polynomial there is always

_{}_{} *f* ( *x* )
_{}

very

_{}_{}*x* _{}

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