# Python Library for Polynomial Arithmetic

Write a Python program to create a Python library for working with polynomial arithmetic.

The task involves creating a Python library to handle polynomial arithmetic. This library should support the creation and manipulation of polynomials, including operations such as addition, subtraction, multiplication, and differentiation. Additionally, it should provide methods for evaluating polynomial expressions at given values. The goal is to enable users to perform these polynomial operations easily and efficiently through a well-defined API.

**Sample Solution**:

**Python Code :**

```
# Define a class for Polynomial arithmetic
class Polynomial:
# Initialize the polynomial with a list of coefficients
def __init__(self, coefficients):
# Coefficients are ordered from the constant term to the highest degree
self.coefficients = coefficients
# Return a string representation of the polynomial for debugging
def __repr__(self):
return "Polynomial(" + ", ".join(map(str, self.coefficients)) + ")"
# Return a human-readable string representation of the polynomial
def __str__(self):
# Create a list to hold terms of the polynomial
terms = []
# Iterate over the coefficients and their indices
for i, coef in enumerate(self.coefficients):
# Skip zero coefficients
if coef != 0:
# Add appropriate string for the term based on its degree
if i == 0:
terms.append(str(coef))
elif i == 1:
terms.append(f"{coef}x")
else:
terms.append(f"{coef}x^{i}")
# Join the terms with ' + ' and return, or return "0" if no terms
return " + ".join(terms) if terms else "0"
# Add two polynomials
def __add__(self, other):
# Determine the maximum length of the two polynomials
max_len = max(len(self.coefficients), len(other.coefficients))
# Initialize the result with zeros
result = [0] * max_len
# Add coefficients of the same degree
for i in range(max_len):
if i < len(self.coefficients):
result[i] += self.coefficients[i]
if i < len(other.coefficients):
result[i] += other.coefficients[i]
# Return a new Polynomial object with the result
return Polynomial(result)
# Subtract one polynomial from another
def __sub__(self, other):
# Determine the maximum length of the two polynomials
max_len = max(len(self.coefficients), len(other.coefficients))
# Initialize the result with zeros
result = [0] * max_len
# Subtract coefficients of the same degree
for i in range(max_len):
if i < len(self.coefficients):
result[i] += self.coefficients[i]
if i < len(other.coefficients):
result[i] -= other.coefficients[i]
# Return a new Polynomial object with the result
return Polynomial(result)
# Multiply two polynomials
def __mul__(self, other):
# Initialize the result with zeros, size is sum of lengths minus 1
result = [0] * (len(self.coefficients) + len(other.coefficients) - 1)
# Perform polynomial multiplication
for i, a in enumerate(self.coefficients):
for j, b in enumerate(other.coefficients):
result[i + j] += a * b
# Return a new Polynomial object with the result
return Polynomial(result)
# Check if two polynomials are equal
def __eq__(self, other):
# Compare coefficients of both polynomials
return self.coefficients == other.coefficients
# Differentiate the polynomial
def differentiate(self):
# If the polynomial is a constant, return zero polynomial
if len(self.coefficients) == 1:
return Polynomial([0])
# Compute the derivative coefficients
result = [i * coef for i, coef in enumerate(self.coefficients)][1:]
# Return a new Polynomial object with the result
return Polynomial(result)
# Evaluate the polynomial at a given value of x
def evaluate(self, x):
# Compute the polynomial value by summing up the terms
return sum(coef * (x ** i) for i, coef in enumerate(self.coefficients))
# Example usage:
if __name__ == "__main__":
# Create polynomial p1: 1 + 2x + 3x^2
p1 = Polynomial([1, 2, 3])
# Create polynomial p2: 3 + 4x
p2 = Polynomial([3, 4])
# Print polynomial p1
print("P1:", p1)
# Print polynomial p2
print("P2:", p2)
# Print the result of adding p1 and p2
print("P1 + P2:", p1 + p2)
# Print the result of subtracting p2 from p1
print("P1 - P2:", p1 - p2)
# Print the result of multiplying p1 and p2
print("P1 * P2:", p1 * p2)
# Check and print if p1 and p2 are equal
print("P1 == P2:", p1 == p2)
# Print the derivative of p1
print("P1 differentiated:", p1.differentiate())
# Print the value of p2 evaluated at x=2
print("P2 evaluated at x=2:", p2.evaluate(2))
```

Output:

P1: 1 + 2x + 3x^2 P2: 3 + 4x P1 + P2: 4 + 6x + 3x^2 P1 - P2: -2 + -2x + 3x^2 P1 * P2: 3 + 10x + 17x^2 + 12x^3 P1 == P2: False P1 differentiated: 2 + 6x P2 evaluated at x=2: 11

**Explanation:**

- Class Initialization (__init__):
- coefficients: List of coefficients starting from the constant term.
- String Representations (__repr__ and str):
- repr: Provides an unambiguous string representation.
- str: Provides a human-readable string representation of the polynomial.
- Addition (__add__):
- Adds two polynomials by summing their coefficients.
- Subtraction (__sub__):
- Subtracts one polynomial from another by subtracting their coefficients.
- Multiplication (__mul__):
- Multiplies two polynomials using nested loops.
- Equality Check (__eq__):
- Checks if two polynomials are equal by comparing their coefficients.
- Differentiation (differentiate):
- Differentiates the polynomial.
- Evaluation (evaluate):
- Evaluates the polynomial at a given value of 'x'.

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