OPERATIONS WITH POLYNOMIALSInoperatingwithpolynomials, the assumption is that the usual laws of the arithmetic
of numbers hold. In arithmetic, the numbers used are the set of real numbers. These consist of the rational numbers and the irrational numbers. Rational numbers are those that can be represented as the ratio of two integers; they include such fractions as y, , and so on, as well as the integers themselves (including 0), and the negatives of all these. The irrational numbers are those that cannot be so represented; they include such numbers as Ã, which require an infinite sequence of digits to be written out as a decimal. These too include negative as well as positive values. Arithmetic alone cannot go beyond the real numbers, but algebra and geometry can include complex numbers.LAWS OF ADDITIONA1.Thesumofanytwo real numbers a and b is again a real number, denoted a + b. The real numbers are closed under the operations of addition, subtraction, multiplication, division, and the extraction of roots; this means that applying any of these operations to real numbers yields a quantity that also is a real number.A2.Nomatterhowterms are grouped in carrying out additions, the sum will always be the same: (a + b) + c = a + (b + c). This is called the associative law of addition.A3.Givenanyrealnumber a, there is a real number zero (0) called the additive identity, such that a + 0 = 0 + a = a.A4.Givenanyrealnumber a, there is a number (-a), called the additive inverse of a, such that (a) + (-a) = 0.A5.Nomatterinwhat order addition is carried out, the sum will always be the same: a + b = b + a. This is called the commutative law of addition.Anysetofnumbersobeying laws A1 to A4 is said to form a group. If the set also obeys A5, it is said to be an Abelian, or commutative, group.LAWS OF MULTIPLICATIONLawssimilartothose for addition also apply to multiplication. Special attention should be given to the multiplicative identity and inverse, M3 and M4.M1.Theproductofany two real numbers a and b is again a real number, denoted a·b or ab.M2.Nomatterhowterms are grouped in carrying out multiplications, the product will always be the same: (ab)c = a(bc). This is called the associative law of multiplication.M3.Givenanyrealnumber a, there is a number one (1) called the multiplicative identity, such that a(1) = 1(a) = a.M4.Givenanynonzero real number a, there is a number (a-1), or (1/a), called the multiplicative inverse, such that a(a-1) = (a-1)a = 1.M5.Nomatterinwhat order multiplication is carried out, the product will always be the same: ab = ba. This is called the commutative law of multiplication.Anysetofelementsobeying these five laws is said to be an Abelian, or commutative, group under multiplication. The set of all real numbers, excluding zero—because division by zero is inadmissible—forms such a commutative group under multiplication.DISTRIBUTIVE LAWSAnotherimportantproperty of the set of real numbers links addition and multiplication in two distributive laws, as follows:D1.
a(
b+
c)=
ab +
acD2.(
b+
c)
a=
ba+
caAnysetofelementswithan equality relation, for which two operations (such as addition and multiplication) are defined, that obeys all the laws for addition A1-A5, the laws for multiplication M1-M5, and the distributive laws D1 and D2, constitutes a field.