Polynomials and properties of Polynomials

Today let us learn about Polynomials and their Properties.

First of all lets try to recollect the meaning of Polynomial.

A Polynomial is a mathematical term used to consist of a sum of terms. Each term may consist a Variable and variables with raised powers: Below is one Sample of a Polynomial:

AX³ + BX² + CX, AX³ + BY²

Now let us see some of the 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), q(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 = 1 . p(x) for any polynomial p(x).