Divide one polynomial by another in Python

To divide one polynomial by another in Python, use the numpy.polynomial.polynomial.polydiv() method. This function performs polynomial division and returns both the quotient and remainder. The arguments are sequences of coefficients from lowest order term to highest, e.g., [1,2,3] represents 1 + 2*x + 3*x².

Syntax

numpy.polynomial.polynomial.polydiv(c1, c2)

Parameters

The parameters are ?

• c1 ? 1-D array of coefficients for the dividend polynomial
• c2 ? 1-D array of coefficients for the divisor polynomial

Return Value

Returns a tuple containing two arrays ?

• Quotient ? Array of coefficients representing the quotient
• Remainder ? Array of coefficients representing the remainder

Example

Let's divide two polynomials and examine the quotient and remainder ?

from numpy.polynomial import polynomial as P

# Define two polynomials
# p1 = 4 + 1*x + 6*x²
# p2 = 2 + 5*x + 3*x²
p1 = (4, 1, 6)
p2 = (2, 5, 3)

print("Polynomial 1 coefficients:", p1)
print("Polynomial 2 coefficients:", p2)

# Divide p1 by p2
quotient, remainder = P.polydiv(p1, p2)
print("\nQuotient:", quotient)
print("Remainder:", remainder)

Polynomial 1 coefficients: (4, 1, 6)
Polynomial 2 coefficients: (2, 5, 3)

Quotient: [2.]
Remainder: [ 0. -9.]

Understanding the Result

The division gives us ?

from numpy.polynomial import polynomial as P

# Define polynomials
p1 = (4, 1, 6)  # 4 + x + 6x²
p2 = (2, 5, 3)  # 2 + 5x + 3x²

quotient, remainder = P.polydiv(p1, p2)

print("Original polynomial p1: 4 + x + 6x²")
print("Divisor polynomial p2: 2 + 5x + 3x²")
print(f"Quotient: {quotient[0]}")
print(f"Remainder: {remainder[1]}x (since remainder[0] = 0)")

# Verification: p1 = quotient * p2 + remainder
verification = P.polyadd(P.polymul(quotient, p2), remainder)
print(f"\nVerification - p1 reconstructed: {verification}")
print(f"Original p1: {p1}")

Original polynomial p1: 4 + x + 6x²
Divisor polynomial p2: 2 + 5x + 3x²
Quotient: 2.0
Remainder: -9.0x (since remainder[0] = 0)

Verification - p1 reconstructed: [ 4.  1.  6.]
Original p1: (4, 1, 6)

Conclusion

The polydiv() function divides polynomials and returns both quotient and remainder as coefficient arrays. This is useful for polynomial long division operations in numerical computing and mathematical analysis.