A scatterplot includes the points (1, 0), (2, 0) , (3, 0), and (0, -6). The data is fitted with a cubic curve whose equation has the form `y=x^(3)+bx^(2)+cx+d`. If the curve passes through all four of the given points, find the value of `b+c`.

Answer

To solve the problem, we need to find the values of \( b \) and \( c \) in the cubic equation \( y = x^3 + bx^2 + cx + d \) such that the curve passes through the points (1, 0), (2, 0), (3, 0), and (0, -6).

### Step 1: Set up the equations
Since the curve passes through the points, we can substitute each point into the equation to create a system of equations.

1. For the point (1, 0):
\[ 0 = 1^3 + b(1^2) + c(1) + d \implies 0 = 1 + b + c + d \implies b + c + d = -1 \quad \text{(Equation 1)} \]

2. For the point (2, 0):
\[ 0 = 2^3 + b(2^2) + c(2) + d \implies 0 = 8 + 4b + 2c + d \implies 4b + 2c + d = -8 \quad \text{(Equation 2)} \]

3. For the point (3, 0):
\[ 0 = 3^3 + b(3^2) + c(3) + d \implies 0 = 27 + 9b + 3c + d \implies 9b + 3c + d = -27 \quad \text{(Equation 3)} \]

4. For the point (0, -6):
\[ -6 = 0^3 + b(0^2) + c(0) + d \implies d = -6 \quad \text{(Equation 4)} \]

### Step 2: Substitute \( d \) into the equations
Now we can substitute \( d = -6 \) into Equations 1, 2, and 3.

1. Substituting into Equation 1:
\[ b + c - 6 = -1 \implies b + c = 5 \quad \text{(Equation 5)} \]

2. Substituting into Equation 2:
\[ 4b + 2c - 6 = -8 \implies 4b + 2c = -2 \implies 2b + c = -1 \quad \text{(Equation 6)} \]

3. Substituting into Equation 3:
\[ 9b + 3c - 6 = -27 \implies 9b + 3c = -21 \implies 3b + c = -7 \quad \text{(Equation 7)} \]

### Step 3: Solve the system of equations
Now we have two equations (Equation 6 and Equation 7) to solve for \( b \) and \( c \).

From Equation 6:
\[ c = -1 - 2b \quad \text{(Equation 8)} \]

Substituting Equation 8 into Equation 7:
\[ 3b + (-1 - 2b) = -7 \]
\[ 3b - 1 - 2b = -7 \]
\[ b - 1 = -7 \implies b = -6 \]

Now substituting \( b = -6 \) back into Equation 8:
\[ c = -1 - 2(-6) = -1 + 12 = 11 \]

### Step 4: Find \( b + c \)
Now we can find \( b + c \):
\[ b + c = -6 + 11 = 5 \]

### Final Answer
Thus, the value of \( b + c \) is \( \boxed{5} \).