Roots of Polynomials

Nerd Cafe | نرد کافه

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 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:

Concept	Python Function
Complex number	complex(a, b)
Real part	z.real
Imaginary part	z.imag
Conjugate	z.conjugate()
Modulus	abs(z)
Phase/angle	cmath.phase(z)
Polar form	cmath.polar(z)
Rectangular form	cmath.rect(r, θ)
Square root	cmath.sqrt(z)
Plotting	matplotlib

Video Tutorial

Support Our Work

If you find this post helpful and would like to support my work, you can send a donation via TRC-20 (USDT). Your contributions help us keep creating and sharing more valuable content.

TRC-20 Address: TAAVVf9ZxUpbyvTa6Gd5SGPmctBdy4PQwf

Thank you for your generosity! 🙏

Keywords

polynomial roots, find roots, numpy roots, sympy solve, quadratic roots, cubic polynomial, complex roots, symbolic computation, numerical roots, polynomial equations, solve polynomial, root finding python, plot polynomial, polynomial graph, rational root theorem, degree of polynomial, polynomial solver, polynomial analysis, math with python, algebraic roots, nerd cafe , نرد کافه

Channel Overview

🌐 Website: www.nerd-cafe.ir

📺 YouTube: @nerd-cafe

🎥 Aparat: nerd_cafe

📌 Pinterest: nerd_cafe

📱 Telegram: @nerd_cafe

📝 Blog: Nerd Café on Virgool

💻 GitHub: nerd-cafe