What Is Polynomials in Data Structure?
In the field of data structures, polynomials are a crucial concept. A polynomial is an algebraic expression that consists of variables, coefficients, and exponents. It is used to represent a mathematical function or equation.
A polynomial can be represented using different data structures. One common approach is to use an array to store the coefficients of each term in the polynomial. The index of the array represents the exponent of the corresponding term.
For example, consider the polynomial:
P(x) = 3x2 + 5x + 2
We can represent this polynomial using an array:
- Index 0: coefficient 2 (constant term)
- Index 1: coefficient 5 (coefficient of x)
- Index 2: coefficient 3 (coefficient of x2)
Addition and Subtraction of Polynomials
To perform addition or subtraction on polynomials, we add or subtract corresponding terms with the same exponent. This operation can be implemented efficiently using arrays.
To add two polynomials, we simply add their corresponding coefficients:
- P1(x) = x3 + 4x2 + x – 6
- P2(x) = -3x3 + x2
Adding these polynomials:
P1(x) + P2(x) = (-3x3 + x2) + (x3 + 4x2 + x – 6)
We combine the terms with the same exponent:
- x3: -3 + 1 = -2x3
- x2: 1 + 4 = 5x2
- x: 0 (no common term)
- x0: -6 (constant term)
The result is: -2x3+5x2-6
Multiplication of Polynomials
Multiplication of polynomials is more complex than addition or subtraction. To multiply two polynomials, we multiply each term of one polynomial with every term of the other polynomial and then combine like terms.
P1(x) = x+1, P2(x) = x+2
Multiplying these polynomials:
P1(x) * P2(x) = (x+1)(x+2)
We use the distributive property to expand the expression:
- (x * x) + (x * 2) + (1 * x) + (1 * 2)
- x2 + 2x + x + 2
The result is: x2 + 3x + 2
Polynomials are an essential concept in data structures. They allow us to represent mathematical functions or equations efficiently.
By using arrays to store coefficients, we can perform operations like addition, subtraction, and multiplication on polynomials. Understanding polynomials is crucial for solving many real-world problems in various fields such as mathematics, computer science, and engineering.
-  Data Structures and Algorithms in Python by Michael T. Goodrich et al.
-  Introduction to Algorithms by Thomas H. Cormen et al.