Operations in the Set of Polynomials

Introduction to Polynomials as a Mathematical Structure

The set of all polynomials with real coefficients forms an important algebraic structure that allows for performing basic arithmetic operations. Every polynomial can be written as:

P(x) = a0 + a1x + a2x^2 + ... + anx^n,

where ai ∈ R and n ∈ N0. These expressions can be added, subtracted, multiplied, and divided by other polynomials. The result of each of these operations (except division with remainder) is again a polynomial.

Addition and Subtraction of Polynomials

Polynomials are added or subtracted term by term, meaning we combine coefficients with the same power of the variable.

Example:
P(x) = 2x^2 + 3x - 1
Q(x) = x^2 - 5x + 4

P(x) + Q(x) = (2x^2 + x^2) + (3x - 5x) + (-1 + 4) = 3x^2 - 2x + 3

Subtraction works the same way, considering the change of signs.

Multiplication of Polynomials

Multiplication follows the distributive law: each term of the first polynomial is multiplied by each term of the second, and then the result is simplified.

Example:
P(x) = x + 2
Q(x) = x - 3

P(x)Q(x) = xx + x*(-3) + 2x + 2(-3) = x^2 - x - 6

In the set of polynomials with real coefficients, the result is always a new polynomial.

Division of Polynomials

Dividing one polynomial by another (of lower or equal degree) works similarly to number division, using the process of long division with remainder.

Example:
Divide P(x) = x^3 + 2x^2 - x - 2 by D(x) = x - 1

By performing the division, we get:

P(x) = (x - 1)*(x^2 + 3x + 2) + 0
Result: the quotient is x^2 + 3x + 2, and the remainder is 0.

If the remainder is not zero, it can be written as a fraction:
P(x)/D(x) = quotient + (remainder / D(x))

Multiplicative Neutrality and Commutativity

For polynomials, the following properties hold:

• Addition and multiplication are commutative and associative operations.
• There exists a neutral element for addition (the zero polynomial) and for multiplication (the polynomial 1).
• Multiplication is distributive over addition.

Conclusion

Operations in the set of polynomials are basic algebraic operations performed within a unified mathematical structure. These operations follow the properties of numerical systems and form the foundation for further studies of equations, functions, and algebraic structures. Each operation, from addition to division, preserves the structure and systematic nature characteristic of polynomials.