The problem requires finding the coefficients $a, b, c, d$ of the cubic polynomial $P(x) = a + bx + cx^2 + dx^3$ that interpolates the given data points: $(-1,1)$, $(0,3)$, $(1, 2)$, and $(2, 4)$.

Polynomial Interpolation: Forming Equations

By substituting the data points into the polynomial equation $P(x) = a + bx + cx^2 + dx^3$, we obtain a system of linear equations:

• For $(-1,1)$: $a + b(-1) + c(-1)^2 + d(-1)^3 = 1 \implies a - b + c - d = 1$
• For $(0,3)$: $a + b(0) + c(0)^2 + d(0)^3 = 3 \implies a = 3$
• For $(1,2)$: $a + b(1) + c(1)^2 + d(1)^3 = 2 \implies a + b + c + d = 2$
• For $(2,4)$: $a + b(2) + c(2)^2 + d(2)^3 = 4 \implies a + 2b + 4c + 8d = 4$

Interpolation: Solving for Coefficients

We use the value $a = 3$ obtained from the point $(0,3)$ and substitute it into the other three equations:

1. Substituting $a=3$ into $a - b + c - d = 1$: $3 - b + c - d = 1 \implies -b + c - d = -2$ (Equation 1)
2. Substituting $a=3$ into $a + b + c + d = 2$: $3 + b + c + d = 2 \implies b + c + d = -1$ (Equation 2)
3. Substituting $a=3$ into $a + 2b + 4c + 8d = 4$: $3 + 2b + 4c + 8d = 4 \implies 2b + 4c + 8d = 1$ (Equation 3)

Now, we solve this reduced system:

• Add Equation 1 and Equation 2: $(-b + c - d) + (b + c + d) = -2 + (-1)$ $2c = -3 \implies c = -3/2$
• Substitute $c = -3/2$ into Equation 2: $b + (-3/2) + d = -1$ $b + d = -1 + 3/2 \implies b + d = 1/2$ (Equation 4)
• Substitute $c = -3/2$ into Equation 3: $2b + 4(-3/2) + 8d = 1$ $2b - 6 + 8d = 1$ $2b + 8d = 7$ (Equation 5)

Solve the system formed by Equation 4 and Equation 5:

• From Equation 4, $d = 1/2 - b$. Substitute this into Equation 5: $2b + 8(1/2 - b) = 7$ $2b + 4 - 8b = 7$ $-6b = 3 \implies b = -1/2$
• Substitute $b = -1/2$ back into $d = 1/2 - b$: $d = 1/2 - (-1/2) = 1/2 + 1/2 = 1$

The coefficients are $a=3$, $b=-1/2$, $c=-3/2$, and $d=1$.

Verification: Checking Coefficient Relationships

We now check the relationships presented in the options using the calculated coefficients.

Let's evaluate the statement $a = -2c$ and $d = -2b$:

• Check $a = -2c$: Is $3 = -2 \times (-3/2)$? $3 = 3$. This is true.
• Check $d = -2b$: Is $1 = -2 \times (-1/2)$? $1 = 1$. This is also true.

Since both conditions $a = -2c$ and $d = -2b$ are satisfied, this statement is correct.