How to find roots of polynomial in Python

In this article, we are going to write the code to find the roots of the Polynomial in Python. Before that, we will learn some basic theories about Polynomial which will help us to understand the codes.

Polynomial: Polynomial comes from two words poly means “many” and nomial means “terms”.

Polynomials are the combinations of variable (x,y,z,etc.), constant(1, 2,-29,1/2,etc.), exponent i.e power of variable (such as 5 in x^5 etc. but only 0,1,2,…) which are combined by addition, subtraction, multiplication, division, except not division by a variable (like 2/x).

Let us consider an example, x+15 in this ‘x’ is called variable.
Polynomial

Power of ‘x’, i.e. 2 is called the exponent/order/degree.
Multiple of ‘x’, i.e. 1,2 is called the coefficient.
The term ‘2’ is called constant.
Monomials items i.e x^2, 2x, 15 are called terms.
How to find roots of polynomial in Python

Roots Of The Polynomials: Roots of the polynomials are defined as the values of the variable which evaluates the polynomials to the zero.
Roots Of The Polynomials
Degree Of The Polynomial: The degree of the polynomial is defined as the highest power of the variable of a polynomial.

To find the roots of a polynomial in math, we use the formula. Let’s learn with an example,
Let consider the polynomial, ax^2+bx+c.  The roots of this equation is,
ax^2+bx+c.

Finding The Roots Of The Polynomial in Python

Program to find the roots of the polynomial, x^2+2x+3. We can find the roots, co-efficient, highest order of the polynomial, changing the variable of the polynomial using numpy module in python.

Steps:
step 1:
line 1, Importing the numpy module as np.
step 2: line 3, Storing the polynomial co-efficient in variable ‘p’.
step 3:  line 5, Printing the polynomial with the highest order.

import numpy as np

p = np.poly1d([1, 2, 3])

print(p)

Output:

   2                             
1 x + 2 x + 3

step 4: line  7, Finding the roots of the polynomial and storing in the variable ‘rootsp’.
step 5: line 9, Printing the roots of the polynomial.

import numpy as np

p = np.poly1d([1, 2, 3])

print(p)

rootsp = p.r

print("\nRoots of Polynomials is :", rootsp)

Output:

   2
1 x + 2 x + 3

Roots of Polynomials is : [-1.+1.41421356j -1.-1.41421356j]

step 6: line 11, Evaluating the polynomial at x=2.

import numpy as np

p = np.poly1d([1, 2, 3])

print(p)

rootsp = p.r

print("\nRoots of Polynomials is :", rootsp)

print("\nEvaluating polynomial at x=2:)", p(2))

Output:

   2
1 x + 2 x + 3

Roots of Polynomials is : [-1.+1.41421356j -1.-1.41421356j]

Evaluating polynomial at x=2: 4.25

step 7: line 13,  Finding the co-efficient of polynomial.

import numpy as np

p = np.poly1d([1, 2, 3])

print(p)

rootsp = p.r

print("\nRoots of Polynomials is :", rootsp)

print("\nEvaluating polynomial at x=2:)", p(2))

print("\nCo-efficient of polynomial:", p.c)

Output:

   2
1 x + 2 x + 3

Roots of Polynomials is : [-1.+1.41421356j -1.-1.41421356j]

Evaluating polynomial at x=2: 4.25

Co-efficient of polynomial: [1 2 3]

step 8: We can also change the variable of the polynomial, which shown inline 3.

import numpy as np

p = np.poly1d([1, 2, 3], variable= 'z')  
                                        
print(p)

rootsp = p.r

print("\nRoots of Polynomials is :", rootsp)


print("\nEvaluating polynomial at x=2:",p(0.5))

print("\nCo-efficient of polynomial:", p.c)

Output:

   2
1 z + 2 z + 3

Roots of Polynomials is : [-1.+1.41421356j -1.-1.41421356j]

Evaluating polynomial at x=2: 4.25

Co-efficient of polynomial: [1 2 3]

Basic Arthematic Operation on Polynomial

#importing the module

import numpy as np
p = np.poly1d([5,4,5,6])
print(p)

# Multiplication of the two polynomials

print("\nmultiplication of the polynomials is :\n", p*p)

# Squaring the polynomials

print("\nSquaring the polynomial:\n", p**2)


#Squaring the individual co-efficient
Csquare = np.square(p)

print("\nSquare of the co-efficient is:\n", Csquare)

Output:

   3     2
5 x + 4 x + 5 x + 6

multiplication of the polynomials is :
    6      5      4       3      2
25 x + 40 x + 66 x + 100 x + 73 x + 60 x + 36

Squaring the polynomial:
  6        5      4       3      2
25 x + 40 x + 66 x + 100 x + 73 x + 60 x + 36

Square of the co-efficient is :
[25 16 25 36]

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