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| import sympy as sp | |
| import numpy as np | |
| import matplotlib.pyplot as plt | |
| x, y = sp.symbols('x y') | |
| def generate_polynomial_template(degree): | |
| terms = [f"a{i}*x^{degree - i}" for i in range(degree)] + [f"a{degree}"] | |
| return " + ".join(terms) + " = 0" | |
| def solve_polynomial(degree, coeff_string): | |
| try: | |
| coeffs = [sp.sympify(s) for s in coeff_string.strip().split()] | |
| if len(coeffs) != degree + 1: | |
| return f"⚠️ Please enter exactly {degree + 1} coefficients.", None, None, "" | |
| poly = sum([coeffs[i] * x**(degree - i) for i in range(degree + 1)]) | |
| simplified = sp.simplify(poly) | |
| # Step-by-step factorization | |
| factored_steps = [] | |
| current_expr = simplified | |
| while True: | |
| factored = sp.factor(current_expr) | |
| if factored == current_expr: | |
| break | |
| factored_steps.append(factored) | |
| current_expr = factored | |
| roots = sp.solve(sp.Eq(simplified, 0), x) | |
| root_display = [] | |
| for i, r in enumerate(roots): | |
| r_simplified = sp.nsimplify(r, rational=True) | |
| root_display.append(f"r_{{{i+1}}} = {sp.latex(r_simplified)}") | |
| steps_output = f"### 🧐 Polynomial Expression\n\n$$ {sp.latex(poly)} = 0 $$\n\n" | |
| steps_output += f"### ✏️ Simplified\n\n$$ {sp.latex(simplified)} = 0 $$\n\n" | |
| if factored_steps: | |
| steps_output += "### 🪜 Step-by-Step Factorization\n\n" | |
| for i, step in enumerate(factored_steps, 1): | |
| steps_output += f"**Step {i}:** $$ {sp.latex(step)} = 0 $$\n\n" | |
| else: | |
| steps_output += "### 🤷 No further factorization possible\n\n" | |
| steps_output += "### 🥮 Roots\n\n$$ " + " \\quad ".join(root_display) + " $$" | |
| # Plotting | |
| f_lambdified = sp.lambdify(x, simplified, modules=["numpy"]) | |
| x_vals = np.linspace(-10, 10, 400) | |
| y_vals = f_lambdified(x_vals) | |
| fig, ax = plt.subplots(figsize=(6, 4)) | |
| ax.plot(x_vals, y_vals, label="Polynomial") | |
| ax.axhline(0, color='black', linewidth=0.5) | |
| ax.axvline(0, color='black', linewidth=0.5) | |
| ax.set_title("📈 Graph of the Polynomial") | |
| ax.set_xlabel("x") | |
| ax.set_ylabel("f(x)") | |
| ax.grid(True) | |
| real_roots = [sp.N(r.evalf()) for r in roots if sp.im(r) == 0] | |
| for r in real_roots: | |
| ax.plot([float(r)], [0], 'ro', label="Real Root") | |
| ax.legend() | |
| return steps_output, fig, "", steps_output | |
| except Exception as e: | |
| return f"❌ Error: {e}", None, "", "" | |
| def solve_linear_system(eq1_str, eq2_str): | |
| try: | |
| eq1 = sp.sympify(eq1_str) | |
| eq2 = sp.sympify(eq2_str) | |
| sol = sp.solve((eq1, eq2), (x, y), dict=True) | |
| steps = "### 🔍 Solving System\n\n" | |
| steps += f"**Equation 1:** $$ {sp.latex(eq1)} $$\n\n" | |
| steps += f"**Equation 2:** $$ {sp.latex(eq2)} $$\n\n" | |
| if sol: | |
| sol = sol[0] | |
| steps += f"**Solution:** $$ x = {sp.latex(sol[x])}, \\quad y = {sp.latex(sol[y])} $$\n\n" | |
| else: | |
| steps += "**❌ No unique solution or inconsistent system**\n" | |
| # Plotting | |
| x_vals = np.linspace(-10, 10, 400) | |
| f1 = sp.solve(eq1, y) | |
| f2 = sp.solve(eq2, y) | |
| fig, ax = plt.subplots(figsize=(6, 4)) | |
| if f1 and f2: | |
| y1_vals = sp.lambdify(x, f1[0], modules=["numpy"])(x_vals) | |
| y2_vals = sp.lambdify(x, f2[0], modules=["numpy"])(x_vals) | |
| ax.plot(x_vals, y1_vals, label="Equation 1") | |
| ax.plot(x_vals, y2_vals, label="Equation 2") | |
| if sol: | |
| px = float(sp.N(sol[x])) | |
| py = float(sp.N(sol[y])) | |
| ax.plot(px, py, 'ro') | |
| ax.annotate(f"({px:.2f}, {py:.2f})", (px, py), textcoords="offset points", xytext=(10, 5), | |
| ha='center', color='red') | |
| ax.axhline(0, color='black', linewidth=0.5) | |
| ax.axvline(0, color='black', linewidth=0.5) | |
| ax.set_title("📉 Graph of the Linear System") | |
| ax.set_xlabel("x") | |
| ax.set_ylabel("y") | |
| ax.grid(True) | |
| ax.legend() | |
| return steps, fig, steps | |
| except Exception as e: | |
| return f"❌ Error: {e}", None, "" | |