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We are tasked with finding a polynomial of the form:

\[
f(x) = ax^3 + bx^2 + cx + d
\]

such that the following conditions hold:
- \( f(0) = -6 \)
- \( f(1) = -6 \)
- \( f(3) = -1 \)
- \( f(4) = -4 \)

### Step 1: Apply the given conditions to form a system of equations

1. **From \( f(0) = -6 \):**
   \[
   a(0)^3 + b(0)^2 + c(0) + d = -6 \quad \Rightarrow \quad d = -6
   \]

2. **From \( f(1) = -6 \):**
   \[
   a(1)^3 + b(1)^2 + c(1) + d = -6 \quad \Rightarrow \quad a + b + c - 6 = -6 \quad \Rightarrow \quad a + b + c = 0
   \]

3. **From \( f(3) = -1 \):**
   \[
   a(3)^3 + b(3)^2 + c(3) + d = -1 \quad \Rightarrow \quad 27a + 9b + 3c - 6 = -1 \quad \Rightarrow \quad 27a + 9b + 3c = 5
   \]

4. **From \( f(4) = -4 \):**
   \[
   a(4)^3 + b(4)^2 + c(4) + d = -4 \quad \Rightarrow \quad 64a + 16b + 4c - 6 = -4 \quad \Rightarrow \quad 64a + 16b + 4c = 2
   \]

### Step 2: Solve the system of equations

We now have the following system of equations:
1. \( a + b + c = 0 \)
2. \( 27a + 9b + 3c = 5 \)
3. \( 64a + 16b + 4c = 2 \)

Let's solve this system to find \( a \), \( b \), and \( c \). I will compute this.The solution to the system of equations is:

\[
a = -\frac{2}{3}, \quad b = \frac{7}{2}, \quad c = -\frac{17}{6}
\]

Thus, the polynomial is:

\[
f(x) = -\frac{2}{3}x^3 + \frac{7}{2}x^2 - \frac{17}{6}x - 6
\]

Let me know if you need further details or explanations! Here are some related questions you might find useful:

1. How do we solve systems of linear equations in general?
2. What are the conditions for a unique solution in a system of equations?
3. How does the degree of a polynomial affect its graph's shape?
4. What happens if you change one of the constraints for the polynomial?
5. How can this method be generalized for higher-degree polynomials?

**Tip:** When solving systems of equations, check if simplifying the equations step-by-step helps identify any common factors.

Find a polynomial of the form f(x) = ax^3 + bx^2 + cx + d such that f(0) = -6, f(1) = -6, f(3) = -1, and f(4) = -4.

Learn how to find a cubic polynomial of the form ax^3 + bx^2 + cx + d given specific values of the function at four points. This involves solving a system of equations to determine the coefficients of the polynomial.

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