Question

Develop class Polynomial. The internal representation of a Polynomial is an array of terms. Each term contains a coefficient and an exponent. The term has the coefficient 2 and the exponent \(4 .\) Develop a complete class containing proper constructor and destructor functions as well as set and get functions. The class should also provide the following overloaded operator capabilities: a. Overload the addition operator \((+)\) to add two Polynomiats. b. Overload the subtraction operator (.) to subtract two Polynomials. c. Overload the assignment operator to assign one Polynomial to another. d. Overload the multiplication operator (*) to multiply two Polynomials. e. Overload the addition assignment operator \((+=),\) subtraction assignment operator \((-=),\) and multiplication assignment operator \(\left(*_{=}\right)\)

Short answer

Create a `Polynomial` class with methods for basic operations, and overload operators for addition, subtraction, multiplication, and assignments.

Step-by-step solution

Define the Polynomial Class Structure

First, define the structure of the `Polynomial` class. This class should contain an array of `Term` objects, where each `Term` has a coefficient and an exponent. You can represent this as a vector or list in C++. Define a constructor to initialize the class, a destructor to clean up resources, and methods for setting and getting terms.

Implement Constructor and Destructor

Create a constructor that initializes the polynomial with default or given terms, and a destructor to handle any necessary cleanup (though for standard containers like vectors, explicit destructors are typically not needed).

Create Set and Get Functions

Implement `setTerm` and `getTerm` functions. `setTerm` should update or add a term in the polynomial, and `getTerm` should return the term at a specified index or return terms based on conditions like returning a term with a specific exponent.

Overload the Addition Operator (+)

Implement `operator+`. This should allow creating a new polynomial by adding corresponding terms (i.e., those with the same exponent) of two polynomials. Any terms that do not have matching exponents should be carried over to the result.

Overload the Subtraction Operator (-)

Implement `operator-`. This is similar to the addition operator but involves subtraction of corresponding terms with the same exponent. Resulting terms from subtracting with zero coefficients can be omitted.

Overload the Multiplication Operator (*)

Define `operator*` to multiply two polynomials. Each term from one polynomial is multiplied by each term of the other. The coefficients are multiplied while the exponents are added when forming new terms.

Overload the Assignment Operator (=)

Implement `operator=` to assign the terms from one polynomial to another. Ensure deep copy semantics, meaning the copies have independent storage.

Overload the Addition Assignment Operator (+=)

Create `operator+=` to add a polynomial to the current instance and assign the result back to it. This operator uses logic similar to `operator+` but updates the current object directly.

Overload the Subtraction Assignment Operator (-=)

Implement `operator-=`. This should subtract a polynomial from the current instance and store the result in the current instance using logic similarly from `operator-`.

Overload the Multiplication Assignment Operator (*=)

Finally, implement `operator*=` to multiply a polynomial with the current instance, updating the current instance with the result directly by invoking logic from `operator*`.

Key concepts

Operator Overloading

Operator overloading is a key feature in C++ programming that allows you to redefine the operations normally performed using operators, for objects of your own classes. In the context of a Polynomial class, operator overloading provides a natural way to manipulate polynomial objects using traditional arithmetic symbols.

Here are some examples of overloaded operators and their significance for a polynomial class:

• Addition "+": Redefines the "+" operator to add two polynomials. This involves aligning terms by their exponents and summing their coefficients.
• Subtraction "-": Redefines the "-" operator to handle polynomial subtraction, making it as easy as regular numeric subtraction.
• Multiplication "*": Allows multiplying two polynomials by handling each combination of terms.
• Assignment "=": Supports copying one polynomial to another, ensuring no shared references to the internal data, crucial for preventing unwanted changes.
• Compound assignments like "+=": These operators allow updating a polynomial directly by adding or subtracting another polynomial to/from it.

Operator overloading in this context not only makes the code for polynomial manipulation more readable and intuitive, but it also aligns object operations with mathematical logic.

Object-Oriented Programming

Object-Oriented Programming (OOP) is a paradigm centered around objects rather than actions, focusing on data rather than logic. This approach allows solutions to be centered around defining objects that represent real-world entities, such as polynomials in mathematics.

OOP principles are reflected in this polynomial implementation as:

• Encapsulation: The Polynomial class encapsulates all properties and behaviors of polynomials, protecting the internal representation and only exposing necessary operations through methods.
• Abstraction: By hiding complex internal operations, such as how polynomials are stored or manipulated, users can interact with simple interfaces like addition and subtraction without needing to know the underlying details.
• Inheritance and Polymorphism: While not directly implemented in this example, these powerful OOP concepts could be used to extend functionality. For example, you might use inheritance to create specialized types of polynomials.

OOP helps in modularizing code, making it more manageable, especially in large-scale applications. It promotes reusability and ease of maintenance.

C++ Classes

C++ classes form the foundation of object-oriented programming in C++. They are like blueprints defining the properties (data members) and behaviors (member functions) a particular object will have.

In the Polynomial exercise, the class structure captures:

• Data Members: Private arrays or vectors that store coefficients and exponents of the polynomial terms.
• Constructors and Destructors: Special functions to initialize and clean up objects. The constructor sets initial values, while the destructor typically handles resource management – though with C++ containers like vectors, explicit destructors are often unnecessary.
• Member Functions: Setters and getters for accessing the data members, as well as overloaded operator functions to handle arithmetic operations.

C++ classes empower you to create complex data models efficiently, focusing on how the data is managed and interacted with. They also provide scope control, allowing certain details to remain hidden from users.

Data Structures

Data structures are essential programming constructs that define the organization, management, and storage of data. They provide a way to handle data in a structured manner so that it can be efficiently accessed and modified.

In our Polynomial class, data structures are used to manage the terms of the polynomial.

• Array or Vector: The terms of the polynomial, each consisting of a coefficient and an exponent, are stored as structured data – typically using arrays or vectors. Vectors are preferable in C++ for dynamically sized containers like polynomials due to their flexible size and ease of use.
• Term Representation: Each term is a structured entity capturing two pieces of data, which can be thought of as a simple form of composite data structure.

Data structures play a critical role in defining how effective and efficient your code will be. Proper data structure choice often determines the complexity of operations like searching, insertion, and deletion, which are common in arithmetic operations on polynomials.