Roots of Polynomials

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What Are Roots of a Polynomial?

A root (or zero) of a polynomial is a value of x that makes the polynomial equal to zero.

If f(x) is a polynomial, then r is a root if:

$$f(r)=0$$

Step 1: Understanding Polynomials

A polynomial of degree n can be written as:

$$P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+...+a_{1}x+a_{0}$$

We are looking for all values of 𝑥 x such that 𝑃(𝑥) =0.

Step 2: Methods for Finding Roots

1. Factoring (for simple polynomials)

$$x^{2}-5x+6=0\Rightarrow (x-2)(x-3)=0\Rightarrow x=2,3$$

2. Quadratic Formula (for degree 2):

$$x=\frac{-b\pm \sqrt{b^{2}-4ac}}{2a}$$

3. Numerical Methods (for degree > 2)

We use computational tools (like NumPy or SymPy) to find approximate or exact roots.

Step 3: Python Implementation

We’ll use both NumPy (numerical roots) and SymPy (symbolic exact roots).

Example 1: Find roots of a cubic polynomial

Polynomial:

$$P(x)=2x^{3}-3x^{2}-11x+6$$

With NumPy (numerical)

import numpy as np

# Coefficients of the polynomial 2x^3 - 3x^2 - 11x + 6
coefficients = [2, -3, -11, 6]

# Find roots
roots = np.roots(coefficients)
print("Roots (numerical):", roots)

Output:

Roots (numerical): [ 3.  -2.   0.5]

With SymPy (exact/symbolic)

import sympy as sp

x = sp.symbols('x')
# Define the polynomial
poly = 2*x**3 - 3*x**2 - 11*x + 6

# Solve the polynomial equation
roots = sp.solve(poly, x)
print("Roots (symbolic):", roots)

Output:

Roots (symbolic): [-2, 1/2, 3]

Example 2: Complex Roots

Polynomial:

$$P(x)=x^{2}+4x+5$$

Using SymPy:

x = sp.symbols('x')
poly = x**2 + 4*x + 5
roots = sp.solve(poly, x)
print("Roots:", roots)

Output:

Roots: [-2 - I, -2 + I]

Summary: When to Use What

Degree	Method	Python Tool
1	Linear Solve	SymPy
2	Quadratic	SymPy or Manual
3+	Numerical	NumPy
Any	Symbolic	SymPy

Plotting the Polynomial

import matplotlib.pyplot as plt
import numpy as np

coefficients = [2, -3, -11, 6]
p = np.poly1d(coefficients)

x_vals = np.linspace(-5, 5, 400)
y_vals = p(x_vals)

plt.plot(x_vals, y_vals, label="P(x)")
plt.axhline(0, color='gray', linestyle='--')
plt.axvline(0, color='gray', linestyle='--')
plt.title("Plot of Polynomial")
plt.xlabel("x")
plt.ylabel("P(x)")
plt.grid(True)
plt.legend()
plt.show()

Output:

Keywords

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