Real-World Problem in Algebra：Polynomial Root Finding in Economics

Problem Statement

A company produces a product whose profit $ P(x) $ (in dollars) as a function of production quantity $ x $ (in units) is modeled by the polynomial:
$$
P(x) = -2x^3 + 15x^2 - 36x + 50
$$

  1. Determine the production quantities $ x $ (roots of $ P(x) = 0 $) where the profit becomes zero (break-even points).
  2. Visualize the profit function $ P(x) $ to show its behavior and the identified roots.

Python Implementation

Here’s the $Python$ code to solve the problem and visualize the results:

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import numpy as np
import matplotlib.pyplot as plt
from numpy.polynomial.polynomial import Polynomial

# Define the profit polynomial P(x) = -2x^3 + 15x^2 - 36x + 50
coefficients = [-2, 15, -36, 50]  # Coefficients in decreasing order of powers
poly = np.poly1d(coefficients)

# Find the roots of the polynomial
roots = np.roots(coefficients)

# Generate data for visualization
x = np.linspace(0, 10, 500)  # Production quantity range
y = poly(x)

# Plotting the polynomial
plt.figure(figsize=(10, 6))
plt.plot(x, y, label="$P(x)$ (Profit Function)", color="blue")

# Highlight the roots
real_roots = roots[np.isreal(roots)].real  # Filter real roots
for root in real_roots:
    plt.scatter(root, 0, color="red", zorder=5, label=f"Root at x = {root:.2f}")

# Add annotations and labels
plt.title("Profit Function $P(x)$ and Break-Even Points", fontsize=14)
plt.axhline(0, color="black", linewidth=0.7, linestyle="--", alpha=0.7)
plt.xlabel("Production Quantity $x$", fontsize=12)
plt.ylabel("Profit $P(x)$", fontsize=12)
plt.legend()
plt.grid(alpha=0.3)

# Show plot
plt.show()

# Print results
print("Roots of P(x):", roots)
print("Real roots (Break-even points):", real_roots)

Explanation of Code

  1. Polynomial Representation:

    • The profit function $ P(x) = -2x^3 + 15x^2 - 36x + 50 $ is represented by its coefficients in decreasing powers of $ x $.
    • The numpy.poly1d function creates a polynomial object for evaluation and visualization.

  2. Root Finding:

    • The roots of $ P(x) $ are found using numpy.roots, which computes all roots (real and complex) of the polynomial.
    • Only real roots are relevant in this context as production quantities $ x $ must be real numbers.

  3. Visualization:

    • The profit function is plotted over a reasonable range of $ x $ values (e.g., $ x \in [0, 10] $).
    • Real roots (break-even points) are highlighted on the graph to show where the profit becomes zero.

Results and Graph Explanation

  1. Numerical Results:

    • Roots of $ P(x) $: These are the solutions to the equation $ P(x) = 0 $.
      Some may be complex, but only real roots are relevant for this real-world context.
    • Real roots (Break-even points): These are production levels where the company neither makes a profit nor a loss.

  2. Graph:

    • The blue curve represents the profit function $ P(x) $.
    • The red points indicate the break-even points where $ P(x) = 0 $.
    • The curve helps visualize how profit changes with production quantity $ x $, showing regions of profit and loss.

By finding and plotting the roots of the polynomial, the company can identify critical production levels to optimize operations.