In this problem, n + 1 data points are given. Find the nth degree polynomial y = f(x) that fits these points. (-3,-17), (-2,3), (2,33), and (3,67)

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:

\[
\begin{aligned}
-27a + 9b - 3c + d &= -17 \\
-8a + 4b - 2c + d &= 3 \\
8a + 4b + 2c + d &= 33 \\
27a + 9b + 3c + d &= 67
\end{aligned}
\]

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!

This problem demonstrates how to find a cubic polynomial that fits four data points: (-3, -17), (-2, 3), (2, 33), and (3, 67). Learn how to set up and solve a system of equations to find the polynomial coefficients. Suitable for Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation