Raise a polynomial to a power in Python

To raise a polynomial to a power in Python, use the numpy.polynomial.polynomial.polypow() method. This function returns the polynomial raised to the specified power, where coefficients are ordered from low to high degree.

Syntax

numpy.polynomial.polynomial.polypow(c, pow, maxpower=16)

Parameters

The function accepts the following parameters:

• c − A 1-D array of polynomial coefficients ordered from low to high degree (e.g., [1,2,3] represents 1 + 2*x + 3*x²)
• pow − The power to which the polynomial will be raised
• maxpower − Maximum power allowed to limit series growth (default is 16)

Example

Let's raise a polynomial (4 + x + 6x²) to the power of 3:

from numpy.polynomial import polynomial as P

# Define polynomial coefficients and power
poly = (4, 1, 6)  # represents 4 + x + 6x²
power = 3

# Display the polynomial
print("Polynomial coefficients:", poly)
print("Power:", power)

# Raise polynomial to the power
result = P.polypow(poly, power)
print("Result coefficients:", result)

Polynomial coefficients: (4, 1, 6)
Power: 3
Result coefficients: [ 64.  48. 300. 145. 450. 108. 216.]

Understanding the Result

The result array represents the coefficients of the expanded polynomial (4 + x + 6x²)³:

from numpy.polynomial import polynomial as P

# Original polynomial
poly = (4, 1, 6)
result = P.polypow(poly, 3)

print("Original polynomial: 4 + x + 6x²")
print("Raised to power 3:")
print("Result: 64 + 48x + 300x² + 145x³ + 450x? + 108x? + 216x?")
print("Coefficients:", result)

Original polynomial: 4 + x + 6x²
Raised to power 3:
Result: 64 + 48x + 300x² + 145x³ + 450x? + 108x? + 216x?
Coefficients: [ 64.  48. 300. 145. 450. 108. 216.]

Using maxpower Parameter

The maxpower parameter prevents excessive growth of polynomial degree:

from numpy.polynomial import polynomial as P

poly = (1, 1, 1)  # 1 + x + x²
high_power = 10

# With default maxpower (16)
result1 = P.polypow(poly, high_power)
print("Degree with default maxpower:", len(result1) - 1)

# With limited maxpower
result2 = P.polypow(poly, high_power, maxpower=8)
print("Degree with maxpower=8:", len(result2) - 1)

Degree with default maxpower: 20
Degree with maxpower=8: 8

Conclusion

The polypow() function efficiently raises polynomials to any power, returning coefficients in ascending degree order. Use the maxpower parameter to control computational complexity for high powers.