Properties of integers
1. The sum of two integers is an integer.
2. Addition of integers is commutative, i.e., a + b = b + a for all integers a, b.
3. Addition of integers is associative, i.e., (a + b) + c = a + (b + c) for all integers a, b, c.
4. The integer zero, (0), is such that a + 0 = a = 0 + a for any integer a.
5. For any integer a, there corresponds an integer -a such that a + (-a) = 0 = (-a) + a.
6. The product of any two integers is an integer.
7. Multiplication of integers is commutative Le., a . b = b . a for any two integers a, b.
8. Multiplication of integers is associative, i.e., (a . b) . c = a . (b . c) for all integers a, b, c.
9. Multiplication 0 integers is distributive over addition, i.e., a .(b + c) = a . b + a. c (b + c) . a = b . a + c . a for all integers a, b, c.
10. The integer 1 is such that a. 1 = a. = 1 . a for any integer a.
Properties of polynomials
1. The sum of two polynomials is a polynomial.
2. Addition of polynomials is commutative i.e., p(x) + q(x) = q(x) + p(x) for all polynomials p(x), q(x).
3. Addition of polynomials is associative, i.e., p(x) + (q(x) + r(x)) = (p(x) + q(x)) + r(x) for all polynomials P(x), r(x).
4. The zero polynomial (0), is such that p(x) + 0 = p(x) = 0 + p(x) for any polynomial p(x).
5. For any polynomial p(x) there corresponds a polynomial -p(x) such that p(x) + [-p(x)) = 0 = [-p(x)) +p(x).
6. The product of any two polynomials is a polynomial.)
7. Multiplication of polynomials is commutative i.e. p(x), q(x) = q(x) . p(x) for any two polynomials p(x). q(x).
8. Multiplication of polynomials is associative i.e. , {p(x) . q(x)] . r(x) = p(x) . {q(x) . r(x)] for all polynomials p(x). q(x). r(x).
9. Multiplication of polynomials is distributive over addition, i.e., p(x) . {q(x)+r(x)} = p(x). q(x) +p(x)r(x), (q(x)+r(x)).p(x)-q(x).p(x) +r(x).p(x) for all polynomials p(x),q(x),r(x).
10. The constant polynomial 1 is such that p(x).1 =p(x) = 1.p(x) for any polynomial p(x)
Thus, We may say that polynomials behave like integers.
Rational expressions: We know that the quotient of two integers is not necessarily an integer i.e., if m and n are integers, n ¹ 0, then the quotient m/n is not necessarily an integer. Therefore, we had to extend our number system and introduced the idea of rational numbers. A rational number is defined as the quotient m/n of two integers m and n, where n ¹ 0. Similarly, if p(x) and q(x) are two polynomials, q(x) ¹ 0, then p(x)/q(x)
need not be a polynomial. So, analogous to the ideal of rational numbers, we define a rational expression as follow:
Rational expression: If p(x) and q(x) are two polynomials, [q(x) non-zero polynomial], then the quotient p(x)/q(x) is called a rational expression.
In the rational expresion p(x)/q(x), p(x) is called the numerator and q(x) is known as the denominator of the rational expression.
Clearly, p(x)/Q(x) need not be a polynomial.
Remark: Since every integer can be written as m/1
Therefore, every integer can be regarded as a rational number. Similarly, every polynomial p(x) can be regarded as a rational expression, since we can write p(x) as p(x)/1.
Thus, every polynomial is a rational expression, but every rational expression need not be a polynomial.
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