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We are tasked with finding values of \(a\), \(b\), \(c\), and \(d\) for the cubic equation \(y = ax^3 + bx^2 + cx + d\) such that it passes through the points \((1, 0)\), \((2, 1)\), and \((3, 2)\).

Let's solve this step by step.

### Step 1: Substituting the points into the equation.

The cubic equation is:

\[
y = ax^3 + bx^2 + cx + d
\]

Substituting the points \((1,0)\), \((2,1)\), and \((3,2)\) will give us a system of linear equations.

1. For the point \((1, 0)\):
   \[
   0 = a(1)^3 + b(1)^2 + c(1) + d \quad \Rightarrow \quad a + b + c + d = 0
   \]
   
2. For the point \((2, 1)\):
   \[
   1 = a(2)^3 + b(2)^2 + c(2) + d \quad \Rightarrow \quad 8a + 4b + 2c + d = 1
   \]
   
3. For the point \((3, 2)\):
   \[
   2 = a(3)^3 + b(3)^2 + c(3) + d \quad \Rightarrow \quad 27a + 9b + 3c + d = 2
   \]

Now, we have the following system of equations:

\[
\begin{align*}
1. & \quad a + b + c + d = 0 \\
2. & \quad 8a + 4b + 2c + d = 1 \\
3. & \quad 27a + 9b + 3c + d = 2
\end{align*}
\]

### Step 2: Solving the system.

We will solve these equations to find \(a\), \(b\), \(c\), and \(d\).The general solution to the system of equations, with respect to \(d\), is as follows:

- \(a = -\frac{d}{6} - \frac{1}{6}\)
- \(b = d + 1\)
- \(c = -\frac{11d}{6} - \frac{5}{6}\)

Since \(d\) is a free parameter, there are infinitely many solutions, depending on the value of \(d\). This means the cubic curve can take different forms depending on the value assigned to \(d\).

Would you like further details on this, or do you have any specific questions?

Here are some additional related questions to consider:
1. What would the curve look like if \(d = 0\)?
2. How does changing \(d\) affect the shape of the cubic curve?
3. What happens if we include a fourth point to constrain the curve further?
4. How does this system demonstrate the flexibility of cubic functions?
5. Can this approach be extended to higher-degree polynomials

Find, if possible, values of a, b, c, d such that the cubic curve y = ax^3 + bx^2 + cx + d passes through the points (1, 0), (2, 1), and (3, 2) in the xy-plane.

This tutorial provides a step-by-step solution to find the coefficients of the cubic equation y = ax^3 + bx^2 + cx + d that passes through the points (1, 0), (2, 1), and (3, 2). Suitable for high school students and involves solving a system of linear equations.

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