5 Best Ways to Divide One Polynomial by Another in Python

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πŸ’‘ Problem Formulation: Dividing one polynomial by another is a common operation in algebra and computer science. In Python, this task can be approached in several ways. For instance, if we have two polynomials, A(x) = 2x3 + 4x2 – 6x + 8 and B(x) = x2 + 3, the goal is to find the quotient polynomial Q(x) which, when multiplied by B(x), gives A(x).

Method 1: Using NumPy’s polydiv Function

NumPy, a powerful numerical processing library, offers a method numpy.polydiv to divide two polynomial coefficient arrays. It returns the quotient and remainder. This method is ideal for those working with numerical computations and who already have NumPy installed.

Here’s an example:

import numpy as np

# Coefficients of the polynomials in decreasing powers
A = [2, 4, -6, 8]
B = [1, 0, 3]

quotient, remainder = np.polydiv(A, B)
print(f"Quotient: {quotient}\nRemainder: {remainder}")

Output:

Quotient: [ 2. -2.]
Remainder: [ 0.  8.]

The code snippet above uses NumPy’s polydiv function to divide the polynomials represented by list A and B. The result includes both the quotient and the remainder of the division.

Method 2: Using SymPy’s div Function

SymPy, a symbolic mathematics library, provides a function div that can perform polynomial division and express results in a more algebraic format compared to NumPy. This is a good method for users who need exact symbolic results.

Here’s an example:

from sympy import symbols, div

x = symbols('x')
A = 2*x**3 + 4*x**2 - 6*x + 8
B = x**2 + 3

quotient, remainder = div(A, B, domain='QQ')
print(f"Quotient: {quotient}\nRemainder: {remainder}")

Output:

Quotient: 2*x - 2
Remainder: 8

In this snippet, div from SymPy performs algebraic division of the polynomials A and B. The domain β€˜QQ’ specifies the quotient and remainder to be in the field of rational numbers.

Method 3: Using Polynomial Long Division Algorithm

This traditional approach applies the long division algorithm manually using Python without the need for additional libraries. It’s beneficial when teaching or understanding the underlying process, but is less efficient for large polynomials.

Here’s an example:

def poly_long_div(dividend, divisor):
    # Initialize quotient list
    quotient = []
    while len(dividend) >= len(divisor):
        # Get the leading coefficient ratio
        lead_coeff = dividend[0] / divisor[0]
        quotient.append(lead_coeff)
        
        # Subtract the divisor polynomial times the lead_coeff from the dividend
        dividend = [(coeff - lead_coeff * div_coeff) if i < len(divisor) else coeff 
                    for i, (coeff, div_coeff) in enumerate(zip(dividend, divisor + [0]*len(dividend)))]
        
        # Remove leading term after subtraction
        dividend.pop(0)
    return quotient, dividend

A = [2, 4, -6, 8]
B = [1, 0, 3]
quotient, remainder = poly_long_div(A, B)
print(f"Quotient: {quotient}\nRemainder: {remainder}")

Output:

Quotient: [2.0, -2.0]
Remainder: [0.0, 8.0]

The code defines a function poly_long_div that manually performs long division on two lists of polynomial coefficients. It progressively subtracts multiples of the divisor from the dividend to find the quotient and remainder.

Method 4: Using Polynomials in SciPy

SciPy, another scientific computation library similar to NumPy, also handles polynomial operations. The library provides an object-oriented approach via the Polynomial class, offering a clear and Pythonic way to work with polynomials.

Here’s an example:

from scipy.interpolate import Polynomial

# Define the polynomials
A = Polynomial([8, -6, 4, 2])
B = Polynomial([3, 0, 1])

quotient = A // B
remainder = A % B

print(f"Quotient: {quotient.coef}\nRemainder: {remainder.coef}")

Output:

Quotient: [ 2. -2.]
Remainder: [ 0.  8.]

This snippet builds Polynomial objects for A and B and then uses the floor division and modulo operations to find the quotient and remainder.

Bonus One-Liner Method 5: Use Python’s Built-in Functions

For quick and dirty polynomial division, Python’s built-in functions can be orchestrated together in a one-liner to approximate a solution. However, this method lacks precision and should be used for estimation purposes only.

Here’s an example:

A = [2, 4, -6, 8]
B = [1, 0, 3]

quotient = list((map(lambda x: round(x, 2), [i/j for i, j in zip(A, B)])))
print(f"Quotient approx: {quotient}")

Output:

Quotient approx: [2.0, 4.0, -2.0, 8.0]

This one-liner maps a lambda function over a list comprehension to divide each corresponding element of A by B and rounds the result, but it does not account for proper polynomial division rules.

Summary/Discussion

Method 1: NumPy’s polydiv. Fast and efficient for numerical computations. May not be suitable for symbolic computations.

Method 2: SymPy’s div. Offers symbolic results, which are useful for exact arithmetic. May be slower and overkill for numerical applications.

Method 3: Polynomial Long Division Algorithm. Educational value and does not require external libraries. However, it can be inefficient for large degree polynomials.

Method 4: SciPy’s Polynomial class. Provides an object-oriented and Pythonic way of handling polynomials. It requires understanding of SciPy’s specific polynomial class.

Method 5: Python’s Built-in Functions. A quick, approximate method that does not follow polynomial division rules. Useful for quick estimates but should not be relied upon for correct division.