**π‘ Problem Formulation:** When working with polynomials in Python, one might encounter the need to perform basic arithmetic operations, such as subtraction. If we have two polynomials, say `P(x) = 3x^2 + 2x + 1`

and `Q(x) = x^2 + x`

, the goal is to find the resulting polynomial after subtraction, which in this case should be `R(x) = 2x^2 + x + 1`

.

## Method 1: Using Numpy’s poly1d Class

The `numpy`

library provides a method for creating polynomial objects using the `poly1d`

class. These polynomial objects can be directly subtracted in an intuitive manner that resembles algebraic operations. This method requires the `numpy`

library to be installed.

Here’s an example:

import numpy as np P = np.poly1d([3, 2, 1]) Q = np.poly1d([1, 1, 0]) R = P - Q print(R)

Output:

2 2 x + 1 x + 1

This code snippet demonstrates the subtraction of two polynomials created with `numpy`

‘s `poly1d`

class. It succinctly performs the subtraction operation using the `-` operator and results in the expected polynomial `R(x)`

.

## Method 2: Using the sympy Library

For symbolic mathematics, the `sympy`

library is an ideal choice. This library provides tools for algebraic operations on polynomials, including subtraction, with exact symbolic representations. The `sympy`

library is particularly useful for higher-level mathematical operations or when an exact symbolic answer is required.

Here’s an example:

from sympy import symbols from sympy import expand x = symbols('x') P = 3*x**2 + 2*x + 1 Q = x**2 + x R = expand(P - Q) print(R)

Output:

2*x**2 + x + 1

This snippet utilizes the `sympy`

library to define polynomials as symbolic expressions and then subtracts them. The `expand`

function is used to ensure the resulting polynomial is simplified.

## Method 3: List-based Coefficient Subtraction

A more manual way of subtracting polynomials is by representing them as lists of coefficients and then subtracting these lists element-wise. This method does not require any external libraries but requires careful handling of the lists, especially when polynomials are of different degrees.

Here’s an example:

P_coef = [3, 2, 1] # Coefficients of P(x) = 3x^2 + 2x + 1 Q_coef = [1, 1, 0] # Coefficients of Q(x) = x^2 + x R_coef = [p - q for p, q in zip(P_coef, Q_coef)] print(R_coef)

Output:

[2, 1, 1]

In this approach, we zip the coefficient lists `P_coef`

and `Q_coef`

together and subtract the elements pairwise, creating a new list `R_coef`

that represents the coefficients of the resulting polynomial.

## Method 4: Using the Polynomial Class of numpy

The `numpy`

library also provides a `Polynomial`

class under its `numpy.polynomial`

module. This class handles polynomial operations and can be a more robust option than using `poly1d`

, especially with future library support in mind.

Here’s an example:

from numpy.polynomial import Polynomial P = Polynomial([1, 2, 3]) # Polynomials are input with coefficients in ascending order Q = Polynomial([0, 1, 1]) R = P - Q print(R)

Output:

Polynomial([ 1., 1., 2.], domain=[-1, 1], window=[-1, 1])

This snippet makes use of numpy’s `Polynomial`

class to handle polynomials as first-class objects. The subtraction utilizes the β-β operator, and the result is printed in a representation that includes coefficients, domain, and window.

## Bonus One-Liner Method 5: Using List Comprehension and zip with Padding

To subtract polynomials of potentially different degrees in one line, you can use list comprehension along with `zip_longest`

from the `itertools`

module to handle the lists of coefficients, padding shorter polynomials with zeros as necessary.

Here’s an example:

from itertools import zip_longest P_coef = [3, 2, 1] Q_coef = [1, 1] R_coef = [p - q for p, q in zip_longest(P_coef, Q_coef, fillvalue=0)] print(R_coef)

Output:

[2, 1, 1]

This compact solution uses list comprehension to subtract element-wise. `zip_longest`

pairs elements, padding with zeroes wherever the lists have different lengths, ensuring proper subtraction without explicit pre-processing of coefficient lists.

## Summary/Discussion

**Method 1:**Numpy’s poly1d. Straightforward and clean syntax. Might not be suitable for symbolic algebra or polynomials with non-integer coefficients.**Method 2:**sympy Library. Provides exact symbolic results. Might be overkill for simple numerical operations and has additional overhead compared to other methods.**Method 3:**List-based Coefficient Subtraction. No external libraries needed. Can be error-prone and requires manual handling of lists of different lengths.**Method 4:**numpy Polynomial Class. Robust and future-proof method within numpy. Slightly more complex syntax due to object’s representation containing additional information.**Method 5:**One-Liner with zip_longest. Quick and concise. Might be less readable for those not familiar with`itertools`

or complex list operations.