typo fix

Signed-off-by: Jonathan Rampersad <jonathan@jono-rams.work>

.all-contributorsrc

@@ -0,0 +1,29 @@
+{
+  "projectName": "PolySolve",
+  "projectOwner": "jono-rams",
+  "repoType": "github",
+  "repoHost": "https://github.com",
+  "files": [
+    "README.md"
+  ],
+  "imageSize": 100,
+  "commit": false,
+  "commitConvention": "angular",
+  "contributorsPerLine": 7,
+  "linkToUsage": true,
+  "skipCi": true,
+  "contributors": [
+    {
+      "login": "jono-rams",
+      "name": "Jonathan Rampersad",
+      "avatar_url": "https://avatars.githubusercontent.com/u/29872001?v=4",
+      "profile": "https://jono-rams.work",
+      "contributions": [
+        "code",
+        "doc",
+        "infra"
+      ]
+    }
+  ],
+  "commitType": "docs"
+}

CONTRIBUTING.md

@@ -1,6 +1,6 @@
 # Contributing to polysolve
 
-First off, thank you for considering contributing! We welcome any contributions, from fixing a typo to implementing a new feature.
+First off, thank you for considering contributing! We welcome any contributions, from fixing a typo to implementing a new feature. You can see a list of everyone who has contributed in our [Contributors Hall of Fame](/.github/CONTRIBUTORS.md).
 
 ## Table of Contents
 
@@ -8,23 +8,26 @@ First off, thank you for considering contributing! We welcome any contributions,
 * [Suggesting Enhancements](#suggesting-enhancements)
 * [Setting Up the Development Environment](#setting-up-the-development-environment)
 * [Running Tests](#running-tests)
+* [Our CI Process & The Gitea Bridge](#our-ci-process--the-gitea-bridge)
 * [Pull Request Process](#pull-request-process)
 
 ## Reporting Bugs
 
-If you find a bug, please open an issue on our Gitea issue tracker. Please include as many details as possible, such as your OS, Python version, steps to reproduce, and any error messages.
+If you find a bug, please open an issue on our GitHub issue tracker. Please include as many details as possible, such as your OS, Python version, steps to reproduce, and any error messages.  
+[Report a Bug](https://github.com/jono-rams/PolySolve/issues/new?assignees=&labels=bug&template=bug_report.md&title=)
 
 ## Suggesting Enhancements
 
-If you have an idea for a new feature or an improvement, please open an issue to discuss it. This allows us to coordinate efforts and ensure the proposed change aligns with the project's goals.
+If you have an idea for a new feature or an improvement, please open an issue to discuss it. This allows us to coordinate efforts and ensure the proposed change aligns with the project's goals.  
+[Suggest an Enhancement](https://github.com/jono-rams/PolySolve/issues/new?assignees=&labels=enhancement&template=feature_request.md&title=)
 
 ## Setting Up the Development Environment
 
-1.  **Fork the repository** on Gitea.
+1.  **Fork the repository** on GitHub.
 
 2.  **Clone your fork** locally:
     ```bash
-    git clone [https://gitea.example.com/YourUsername/PolySolve.git](https://gitea.example.com/YourUsername/PolySolve.git)
+    git clone git clone https://github.com/YourUsername/PolySolve.git
     cd PolySolve
     ```
 
@@ -58,34 +61,33 @@ We use `pytest` for automated testing. After setting up the development environm
 pytest
 ```
 
-This will automatically discover and run all tests located in the `tests/` directory.
+This will automatically discover and run all tests located in the `tests/` directory. All tests should pass before you submit your changes.  
+If you are adding a new feature or fixing a bug, please add a corresponding test case to ensure the code is working correctly and to prevent future regressions.
 
-All tests should pass before you submit your changes. If you are adding a new feature or fixing a bug, please add a corresponding test case to ensure the code is working correctly and to prevent future regressions.
+## Our CI Process & The Gitea Bridge
 
-## CI & Automated Testing Environment
+To ensure that all contributions are consistent and stable, our test suite is executed automatically in a controlled environment. Here’s how it works:
+1. Our canonical source of truth and CI/CD runners are managed on our private Gitea instance.
+2. When you open a Pull Request on GitHub, our "Gitea Bridge" bot automatically mirrors your changes to a corresponding PR on our Gitea instance.
+3. The tests are run using Gitea Actions within our specific, reproducible environment.
+4. The results (success or failure) are then reported back to your GitHub Pull Request via a status check named "Gitea CI Bridge".
 
-To ensure that all contributions are consistent and stable, our test suite is executed automatically via Gitea Actions. Your pull request must pass all these checks before it can be merged.
+Your pull request must pass all these checks before it can be merged.
-
-Our CI environment is the ultimate source of truth. It is built using a custom Docker image to lock down dependencies and guarantee reproducibility.
 
 ### Reference Environment Specification
 
+You can replicate our CI environment exactly to minimize "it works on my machine" issues.
 * **Base OS:** Ubuntu 24.04
 * **CUDA Toolkit:** 12.5.1
 * **Base Docker Image:** `nvidia/cuda:12.5.1-devel-ubuntu24.04`
 * **Node.js Version:** 20.x
 * **Python Versions Tested:** 3.8, 3.10, 3.12
-
+* **CI Docker Image:** You can pull the exact image used by our runners from Docker Hub:
-### Reproducing the Environment
-
-You can replicate our CI environment exactly by using the public Docker image we built for our runners. You can pull it from Docker Hub:
 
 ```bash
 docker pull c1ph3rd3v/gitea-runner-cuda-node:12.5.1-ubuntu24.04
 ```
 
-Using this Docker image for your local development will ensure your code runs in the same environment as our automated tests, minimizing "it works on my machine" issues.
-
 ## Pull Request Process
 
 1.  Create a new branch for your feature or bugfix from the `main` branch:
@@ -99,7 +101,7 @@ Using this Docker image for your local development will ensure your code runs in
     pytest
     ```
 5.  Commit your changes with a clear and descriptive commit message.
-6.  Push your branch to your fork on Gitea.
+6.  Push your branch to your fork on GitHub.
-7.  Open a pull request to the `main` branch of the original `PolySolve` repository. Please provide a clear title and description for your pull request.
+7.  Open a **Pull Request** to the `main` branch of the `jono-rams/PolySolve` repository on **GitHub**. Please provide a clear title and description for your pull request.
 
 Once you submit your pull request, our automated CI tests will run. We will review your contribution and provide feedback as soon as possible. Thank you for your contribution!

README.md

@@ -1,4 +1,6 @@
-# polysolve
+<p align="center">
+  <img src="https://i.ibb.co/N22Gx6xq/Poly-Solve-Logo.png" alt="polysolve Logo" width="256">
+</p>
 
 [![PyPI version](https://img.shields.io/pypi/v/polysolve.svg)](https://pypi.org/project/polysolve/)
 [![PyPI pyversions](https://img.shields.io/pypi/pyversions/polysolve.svg)](https://pypi.org/project/polysolve/)
@@ -9,7 +11,7 @@ A Python library for representing, manipulating, and solving polynomial equation
 
 ## Key Features
 
-* **Create and Manipulate Polynomials**: Easily define polynomials of any degree and perform arithmetic operations like addition, subtraction, and scaling.
+* **Create and Manipulate Polynomials**: Easily define polynomials of any degree using integer or float coefficients, and perform arithmetic operations like addition, subtraction, multiplication, and scaling.
 * **Genetic Algorithm Solver**: Find approximate real roots for complex polynomials where analytical solutions are difficult or impossible.
 * **CUDA Accelerated**: Leverage NVIDIA GPUs for a massive performance boost when finding roots in large solution spaces.
 * **Analytical Solvers**: Includes standard, exact solvers for simple cases (e.g., `quadratic_solve`).
@@ -44,6 +46,7 @@ Here is a simple example of how to define a quadratic function, find its propert
 from polysolve import Function, GA_Options, quadratic_solve
 
 # 1. Define the function f(x) = 2x^2 - 3x - 5
+#    Coefficients can be integers or floats.
 f1 = Function(largest_exponent=2)
 f1.set_constants([2, -3, -5])
 
@@ -56,17 +59,22 @@ print(f"Value of f1 at x=5 is: {y_val}")
 # > Value of f1 at x=5 is: 30.0
 
 # 3. Get the derivative: 4x - 3
-df1 = f1.differential()
+df1 = f1.derivative()
 print(f"Derivative of f1: {df1}")
 # > Derivative of f1: 4x - 3
 
-# 4. Find roots analytically using the quadratic formula
+# 4. Get the 2nd derivative: 4
+ddf1 = f1.nth_derivative(2)
+print(f"2nd Derivative of f1: {ddf1}")
+# > Derivative of f1: 4
+
+# 5. Find roots analytically using the quadratic formula
 #    This is exact and fast for degree-2 polynomials.
 roots_analytic = quadratic_solve(f1)
 print(f"Analytic roots: {sorted(roots_analytic)}")
 # > Analytic roots: [-1.0, 2.5]
 
-# 5. Find roots with the genetic algorithm (CPU)
+# 6. Find roots with the genetic algorithm (CPU)
 #    This can solve polynomials of any degree.
 ga_opts = GA_Options(num_of_generations=20)
 roots_ga = f1.get_real_roots(ga_opts, use_cuda=False)
@@ -89,10 +97,41 @@ While the code may work on other configurations, all contributions must pass the
 
 ## Contributing
 
+[![PRs Welcome](https://img.shields.io/badge/PRs-welcome-brightgreen.svg?style=flat-square)](http://makeapullrequest.com)
+[![GitHub issues](https://img.shields.io/github/issues/jono-rams/PolySolve.svg?style=flat-square)](https://github.com/jono-rams/PolySolve/issues)
+[![GitHub pull requests](https://img.shields.io/github/issues-pr/jono-rams/PolySolve.svg?style=flat-square)](https://github.com/jono-rams/PolySolve/pulls)
+
 Contributions are welcome! Whether it's a bug report, a feature request, or a pull request, please feel free to get involved.
 
 Please read our `CONTRIBUTING.md` file for details on our code of conduct and the process for submitting pull requests.
 
+## Contributors
+
+<!-- ALL-CONTRIBUTORS-LIST:START - Do not remove or modify this section -->
+<!-- prettier-ignore-start -->
+<!-- markdownlint-disable -->
+<table>
+  <tbody>
+    <tr>
+      <td align="center" valign="top" width="14.28%"><a href="https://jono-rams.work"><img src="https://avatars.githubusercontent.com/u/29872001?v=4?s=100" width="100px;" alt="Jonathan Rampersad"/><br /><sub><b>Jonathan Rampersad</b></sub></a><br /><a href="https://github.com/jono-rams/PolySolve/commits?author=jono-rams" title="Code">💻</a> <a href="https://github.com/jono-rams/PolySolve/commits?author=jono-rams" title="Documentation">📖</a> <a href="#infra-jono-rams" title="Infrastructure (Hosting, Build-Tools, etc)">🚇</a></td>
+    </tr>
+  </tbody>
+  <tfoot>
+    <tr>
+      <td align="center" size="13px" colspan="7">
+        <img src="https://raw.githubusercontent.com/all-contributors/all-contributors-cli/1b8533af435da9854653492b1327a23a4dbd0a10/assets/logo-small.svg">
+          <a href="https://all-contributors.js.org/docs/en/bot/usage">Add your contributions</a>
+        </img>
+      </td>
+    </tr>
+  </tfoot>
+</table>
+
+<!-- markdownlint-restore -->
+<!-- prettier-ignore-end -->
+
+<!-- ALL-CONTRIBUTORS-LIST:END -->
+
 ## License
 
 This project is licensed under the MIT License - see the `LICENSE` file for details.

pyproject.toml

@@ -5,7 +5,7 @@ build-backend = "setuptools.build_meta"
 [project]
 # --- Core Metadata ---
 name = "polysolve"
-version = "0.1.0"
+version = "0.3.2"
 authors = [
   { name="Jonathan Rampersad", email="jonathan@jono-rams.work" },
 ]
@@ -42,5 +42,7 @@ cuda12 = ["cupy-cuda12x"]
 dev = ["pytest"]
 
 [project.urls]
-Homepage = "https://gitea.jono-rams.work/jono/PolySolve"
+Homepage = "https://polysolve.jono-rams.work"
-"Bug Tracker" = "https://gitea.jono-rams.work/jono/PolySolve/issues"
+Documentation = "https://polysolve.jono-rams.work/docs"
+Repository  = "https://github.com/jono-rams/PolySolve"
+"Bug Tracker" = "https://github.com/jono-rams/PolySolve/issues"

src/polysolve/__init__.py

@@ -1,7 +1,7 @@
 import math
 import numpy as np
 from dataclasses import dataclass
-from typing import List, Optional
+from typing import List, Optional, Union
 import warnings
 
 # Attempt to import CuPy for CUDA acceleration.
@@ -12,8 +12,32 @@ try:
 except ImportError:
     _CUPY_AVAILABLE = False
 
-# The CUDA kernel for the fitness function
+# The CUDA kernels for the fitness function
-_FITNESS_KERNEL = """
+_FITNESS_KERNEL_FLOAT = """
+extern "C" __global__ void fitness_kernel(
+    const double* coefficients, 
+    int num_coefficients, 
+    const double* x_vals, 
+    double* ranks, 
+    int size, 
+    double y_val)
+{
+    int idx = threadIdx.x + blockIdx.x * blockDim.x;
+    if (idx < size)
+    {
+        double ans = 0;
+        int lrgst_expo = num_coefficients - 1;
+        for (int i = 0; i < num_coefficients; ++i)
+        {
+            ans += coefficients[i] * pow(x_vals[idx], (double)(lrgst_expo - i));
+        }
+
+        ans -= y_val;
+        ranks[idx] = (ans == 0) ? 1.7976931348623157e+308 : fabs(1.0 / ans);
+    }
+}
+"""
+_FITNESS_KERNEL_INT = """
 extern "C" __global__ void fitness_kernel(
     const long long* coefficients, 
     int num_coefficients, 
@@ -38,6 +62,7 @@ extern "C" __global__ void fitness_kernel(
 }
 """
 
+
 @dataclass
 class GA_Options:
     """
@@ -76,13 +101,14 @@ class Function:
         self.coefficients: Optional[np.ndarray] = None
         self._initialized = False
 
-    def set_coeffs(self, coefficients: List[int]):
+    def set_coeffs(self, coefficients: List[Union[int, float]]):
         """
         Sets the coefficients of the polynomial.
 
         Args:
-            coefficients (List[int]): A list of integer coefficients. The list size
+            coefficients (List[Union[int, float]]): A list of integer or float
-                                   must be largest_exponent + 1.
+                                                   coefficients. The list size
+                                                   must be largest_exponent + 1.
 
         Raises:
             ValueError: If the input is invalid.
@@ -95,8 +121,14 @@ class Function:
             )
         if coefficients[0] == 0 and self._largest_exponent > 0:
             raise ValueError("The first constant (for the largest exponent) cannot be 0.")
+        
+        # Check if any coefficient is a float
+        is_float = any(isinstance(c, float) for c in coefficients)
 
-        self.coefficients = np.array(coefficients, dtype=np.int64)
+        # Choose the dtype based on the input
+        target_dtype = np.float64 if is_float else np.int64
+
+        self.coefficients = np.array(coefficients, dtype=target_dtype)
         self._initialized = True
 
     def _check_initialized(self):
@@ -108,6 +140,11 @@ class Function:
     def largest_exponent(self) -> int:
         """Returns the largest exponent of the function."""
         return self._largest_exponent
+    
+    @property
+    def degree(self) -> int:
+        """Returns the largest exponent of the function."""
+        return self._largest_exponent
 
     def solve_y(self, x_val: float) -> float:
         """
@@ -129,15 +166,66 @@ class Function:
         Returns:
             Function: A new Function object representing the derivative.
         """
+        warnings.warn(
+            "The 'differential' function has been renamed. Please use 'derivative' instead.",
+            DeprecationWarning,
+            stacklevel=2
+        )
+
         self._check_initialized()
         if self._largest_exponent == 0:
             raise ValueError("Cannot differentiate a constant (Function of degree 0).")
 
+        return self.derivitive()
+        
+    
+    def derivative(self) -> 'Function':
+        """
+        Calculates the derivative of the function.
+
+        Returns:
+            Function: A new Function object representing the derivative.
+        """
+        self._check_initialized()
+        if self._largest_exponent == 0:
+            raise ValueError("Cannot differentiate a constant (Function of degree 0).")
+        
         derivative_coefficients = np.polyder(self.coefficients)
         
         diff_func = Function(self._largest_exponent - 1)
         diff_func.set_coeffs(derivative_coefficients.tolist())
         return diff_func
+    
+
+    def nth_derivative(self, n: int) -> 'Function':
+        """
+        Calculates the nth derivative of the function.
+
+        Args:
+            n (int): The order of the derivative to calculate.
+
+        Returns:
+           Function: A new Function object representing the nth derivative.
+        """
+        self._check_initialized()
+        
+        if not isinstance(n, int) or n < 1:
+            raise ValueError("Derivative order 'n' must be a positive integer.")
+
+        if n > self.largest_exponent:
+            function = Function(0)
+            function.set_coeffs([0])
+            return function
+
+        if n == 1:
+            return self.derivative()
+        
+        function = self
+        for _ in range(n):
+            function = function.derivative()
+
+        return function
+
 
     def get_real_roots(self, options: GA_Options = GA_Options(), use_cuda: bool = False) -> np.ndarray:
         """
@@ -219,11 +307,16 @@ class Function:
 
     def _solve_x_cuda(self, y_val: float, options: GA_Options) -> np.ndarray:
         """Genetic algorithm implementation using CuPy (GPU/CUDA)."""
-        # Load the raw CUDA kernel
-        fitness_gpu = cupy.RawKernel(_FITNESS_KERNEL, 'fitness_kernel')
         
-        # Move coefficients to GPU
+        # Check the dtype of our coefficients array
-        d_coefficients = cupy.array(self.coefficients, dtype=cupy.int64)
+        if self.coefficients.dtype == np.float64:
+            fitness_gpu = cupy.RawKernel(_FITNESS_KERNEL_FLOAT, 'fitness_kernel')
+            d_coefficients = cupy.array(self.coefficients, dtype=cupy.float64)
+        elif self.coefficients.dtype == np.int64:
+            fitness_gpu = cupy.RawKernel(_FITNESS_KERNEL_INT, 'fitness_kernel')
+            d_coefficients = cupy.array(self.coefficients, dtype=cupy.int64)
+        else:
+            raise TypeError(f"Unsupported dtype for CUDA solver: {self.coefficients.dtype}")
         
         # Create initial random solutions on the GPU
         d_solutions = cupy.random.uniform(
@@ -281,12 +374,16 @@ class Function:
             power = self._largest_exponent - i
             
             # Coefficient part
-            if c == 1 and power != 0:
+            coeff_val = c
+            if c == int(c):
+                coeff_val = int(c)
+
+            if coeff_val == 1 and power != 0:
                 coeff = ""
-            elif c == -1 and power != 0:
+            elif coeff_val == -1 and power != 0:
                 coeff = "-"
             else:
-                coeff = str(c)
+                coeff = str(coeff_val)
 
             # Variable part
             if power == 0:
@@ -300,7 +397,7 @@ class Function:
             sign = ""
             if i > 0:
                 sign = " + " if c > 0 else " - "
-                coeff = str(abs(c))
+                coeff = str(abs(coeff_val))
                 if abs(c) == 1 and power != 0:
                     coeff = "" # Don't show 1 for non-constant terms
 
@@ -336,34 +433,55 @@ class Function:
         result_func = Function(len(new_coefficients) - 1)
         result_func.set_coeffs(new_coefficients.tolist())
         return result_func
-        
+    
-    def __mul__(self, scalar: int) -> 'Function':
+    def _multiply_by_scalar(self, scalar: Union[int, float]) -> 'Function':
-        """Multiplies the function by a scalar constant."""
+        """Helper method to multiply the function by a scalar constant."""
-        self._check_initialized()
+        self._check_initialized() # It's good practice to check here too
-        if not isinstance(scalar, (int, float)):
-            return NotImplemented
-        if scalar == 0:
-            raise ValueError("Cannot multiply a function by 0.")
 
+        if scalar == 0:
+            result_func = Function(0)
+            result_func.set_coeffs([0])
+            return result_func
+    
         new_coefficients = self.coefficients * scalar
-        
+    
         result_func = Function(self._largest_exponent)
         result_func.set_coeffs(new_coefficients.tolist())
         return result_func
 
-    def __rmul__(self, scalar: int) -> 'Function':
+    def _multiply_by_function(self, other: 'Function') -> 'Function':
+        """Helper method for polynomial multiplication (Function * Function)."""
+        self._check_initialized()
+        other._check_initialized()
+
+        # np.polymul performs convolution of coefficients to multiply polynomials
+        new_coefficients = np.polymul(self.coefficients, other.coefficients)
+    
+        # The degree of the resulting polynomial is derived from the new coefficients
+        new_degree = len(new_coefficients) - 1
+    
+        result_func = Function(new_degree)
+        result_func.set_coeffs(new_coefficients.tolist())
+        return result_func
+        
+    def __mul__(self, other: Union['Function', int, float]) -> 'Function':
+        """Multiplies the function by a scalar constant."""
+        if isinstance(other, (int, float)):
+            return self._multiply_by_scalar(other)
+        elif isinstance(other, self.__class__):
+            return self._multiply_by_function(other)
+        else:
+            return NotImplemented
+
+    def __rmul__(self, scalar: Union[int, float]) -> 'Function':
         """Handles scalar multiplication from the right (e.g., 3 * func)."""
+
         return self.__mul__(scalar)
         
-    def __imul__(self, scalar: int) -> 'Function':
+    def __imul__(self, other: Union['Function', int, float]) -> 'Function':
         """Performs in-place multiplication by a scalar (func *= 3)."""
-        self._check_initialized()
+
-        if not isinstance(scalar, (int, float)):
+        self.coefficients *= other
-            return NotImplemented
-        if scalar == 0:
-            raise ValueError("Cannot multiply a function by 0.")
-        
-        self.coefficients *= scalar
         return self
 
 
@@ -408,9 +526,13 @@ if __name__ == '__main__':
     print(f"Value of f1 at x=5 is: {y}") # Expected: 2*(25) - 3*(5) - 5 = 50 - 15 - 5 = 30
 
     # Find the derivative: 4x - 3
-    df1 = f1.differential()
+    df1 = f1.derivative()
     print(f"Derivative of f1: {df1}")
 
+    # Find the second derivative: 4
+    ddf1 = f1.nth_derivative(2)
+    print(f"Second derivative of f1: {ddf1}")
+
     # --- Root Finding ---
     # 1. Analytical solution for quadratic
     roots_analytic = quadratic_solve(f1)
@@ -448,4 +570,18 @@ if __name__ == '__main__':
 
     # Multiplication: (x + 10) * 3 = 3x + 30
     f_mul = f2 * 3
-    print(f"f2 * 3 = {f_mul}")
+    print(f"f2 * 3 = {f_mul}")
+
+    # f3 represents 2x^2 + 3x + 1
+    f3 = Function(2)
+    f3.set_coeffs([2, 3, 1]) 
+    print(f"Function f3: {f3}")
+
+    # f4 represents 5x - 4
+    f4 = Function(1)
+    f4.set_coeffs([5, -4])
+    print(f"Function f4: {f4}")
+
+    # Multiply the two functions
+    product_func = f3 * f4
+    print(f"f3 * f4 = {product_func}")

tests/test_polysolve.py

@@ -24,6 +24,18 @@ def linear_func() -> Function:
     f.set_coeffs([1, 10])
     return f
 
+@pytest.fixture
+def m_func_1() -> Function:
+    f = Function(2)
+    f.set_coeffs([2, 3, 1])
+    return f
+
+@pytest.fixture
+def m_func_2() -> Function:
+    f = Function(1)
+    f.set_coeffs([5, -4])
+    return f
+
 # --- Core Functionality Tests ---
 
 def test_solve_y(quadratic_func):
@@ -32,13 +44,20 @@ def test_solve_y(quadratic_func):
     assert quadratic_func.solve_y(0) == -5.0
     assert quadratic_func.solve_y(-1) == 0.0
 
-def test_differential(quadratic_func):
+def test_derivative(quadratic_func):
     """Tests the calculation of the function's derivative."""
-    derivative = quadratic_func.differential()
+    derivative = quadratic_func.derivative()
     assert derivative.largest_exponent == 1
     # The derivative of 2x^2 - 3x - 5 is 4x - 3
     assert np.array_equal(derivative.coefficients, [4, -3])
 
+def test_nth_derivative(quadratic_func):
+    """Tests the calculation of the function's 2nd derivative."""
+    derivative = quadratic_func.nth_derivative(2)
+    assert derivative.largest_exponent == 0
+    # The derivative of 2x^2 - 3x - 5 is 4x - 3
+    assert np.array_equal(derivative.coefficients, [4])
+
 def test_quadratic_solve(quadratic_func):
     """Tests the analytical quadratic solver for exact roots."""
     roots = quadratic_solve(quadratic_func)
@@ -61,13 +80,20 @@ def test_subtraction(quadratic_func, linear_func):
     assert result.largest_exponent == 2
     assert np.array_equal(result.coefficients, [2, -4, -15])
 
-def test_multiplication(linear_func):
+def test_scalar_multiplication(linear_func):
     """Tests the multiplication of a Function object by a scalar."""
     # (x + 10) * 3 = 3x + 30
     result = linear_func * 3
     assert result.largest_exponent == 1
     assert np.array_equal(result.coefficients, [3, 30])
 
+def test_function_multiplication(m_func_1, m_func_2):
+    """Tests the multiplication of two Function objects."""
+    # (2x^2 + 3x + 1) * (5x -4) = 10x^3 + 7x^2 - 7x -4
+    result = m_func_1 * m_func_2
+    assert result.largest_exponent == 3
+    assert np.array_equal(result.coefficients, [10, 7, -7, -4])
+
 # --- Genetic Algorithm Root-Finding Tests ---
 
 def test_get_real_roots_numpy(quadratic_func):