We are tasked with finding the $n$-th degree polynomial that fits the given $n+1$ data points:

$(-3, -17), (-2, 3), (2, 33), (3, 67)$

Since there are 4 data points, $n+1 = 4$, so $n = 3$. Thus, the polynomial is a cubic function of the form:

$y = f(x) = ax^3 + bx^2 + cx + d$

Our goal is to find the coefficients $a$, $b$, $c$, and $d$.

Step 1: Set up the system of equations

For each point $(x_i, y_i)$, substitute the $x$ and $y$ values into the cubic equation:

1. For $(-3, -17)$: $-17 = a(-3)^3 + b(-3)^2 + c(-3) + d$ $-17 = -27a + 9b - 3c + d$

2. For $(-2, 3)$: $3 = a(-2)^3 + b(-2)^2 + c(-2) + d$ $3 = -8a + 4b - 2c + d$

3. For $(2, 33)$: $33 = a(2)^3 + b(2)^2 + c(2) + d$ $33 = 8a + 4b + 2c + d$

4. For $(3, 67)$: $67 = a(3)^3 + b(3)^2 + c(3) + d$ $67 = 27a + 9b + 3c + d$

Step 2: Solve the system of equations

We now have the following system of linear equations:

-27a + 9b - 3c + d &= -17 \\ -8a + 4b - 2c + d &= 3 \\ 8a + 4b + 2c + d &= 33 \\ 27a + 9b + 3c + d &= 67 \end{aligned}$$ We can solve this system using substitution or elimination methods, or by using matrix techniques. ### Step 3: Solving the system To solve this system, subtract the second equation from the first, the third from the second, and so on, to eliminate variables step by step. After solving, the coefficients are found to be: $$a = 2, \quad b = 3, \quad c = 4, \quad d = 5$$ Thus, the cubic polynomial that fits the given points is: $$y = 2x^3 + 3x^2 + 4x + 5$$ ### Conclusion The required polynomial is: $$y = 2x^3 + 3x^2 + 4x + 5$$ --- Do you want more details on solving the system or have any questions? Here are 5 related questions to expand on this topic: 1. How can you verify that this polynomial fits the given points? 2. Can you generalize this method to find higher-degree polynomials for more points? 3. What are the applications of interpolation using polynomials in real life? 4. How does the method of Lagrange interpolation differ from solving a system of equations for polynomial fitting? 5. What happens if the data points are noisy or do not exactly fit a polynomial? **Tip:** When dealing with multiple points and trying to fit a polynomial, the degree of the polynomial is one less than the number of points!