Finding a polynomial when only given five points

In summary, the conversation revolves around a student who is taking a linear algebra class and has been given a calculator project with four linear algebra application problems. The first problem involves determining a polynomial that passes through given points. The second problem requires sketching the graph of the polynomial. The student is unsure how to approach the first problem and has asked the professor for help. They have been told to set up a system of linear equations and solve it using a program. The student is concerned about the high powers involved in the equation, but the professor reassures them that they are just numbers.
  • #1
cougarsoccer
12
0
i just enrolled at the linear algebra class at my university. and after out first test the professor gave us this "calculator project" (meaning just using the calculator in some way to get the answer) that has 4 linear algebra application problems. i figured out the last three but i am stuck on the first one. it is in two parts.

a. determine the polynomial whose graph passes through the points (-1/2, 75/16), (0,6), (2/3, 220/81), (3,48), and (4,210)

b. the second question just says to sketch an accurate graph of the polynomial.

i went and asked the professor the other day how we might go about finding the answer. and she hinted around having a equation in the fourth degree (ax^4+bx^3+cx^2+dx+e if I'm not mistaken) all the other questions involved me making a matrix out of the system of equations and putting them in reduced row echelon form to find the answer so I'm suspecting that it must be done that way for this one too b/c that is all we have learned in the class so far. any helps or hints would be appreciated guys.
 
Physics news on Phys.org
  • #2
this should be moved to some homework forum.

cougar, you can set up a 5x5 system of linear equations that you can solve with any program that does that (Gaussian elimination or square matrix inversion). don't let those powers of 3 or 4 scare you; they get applied to those known x ordinates: {-1/2, 0, 2/3, 3, 4}. they're just numbers. it's the a, b, c, d, e that are your unknowns.
 

1. How do you find a polynomial when only given five points?

The process of finding a polynomial when only given five points involves using the method of finite differences. This method allows you to determine the coefficients of the polynomial by finding the differences between the given points and using those differences to construct the polynomial.

2. What is the purpose of finding a polynomial when only given five points?

Finding a polynomial when only given five points is useful in many areas of science, such as data analysis and curve fitting. It allows you to approximate a relationship between the given data points and make predictions or extrapolations beyond the given data.

3. Can you find a unique polynomial with only five points?

Yes, it is possible to find a unique polynomial with only five points. However, the degree of the polynomial may vary depending on the complexity of the data and the method used to find the polynomial.

4. What are the limitations of finding a polynomial when only given five points?

One limitation is that the resulting polynomial may not accurately represent the true underlying relationship between the data points. This can happen if the data is noisy or if the points are not evenly spaced. Additionally, the degree of the polynomial may need to be high to accurately fit the data, which can lead to overfitting and decreased predictive power.

5. Is there a specific method for finding a polynomial when only given five points?

There are several methods for finding a polynomial when only given five points, such as the method of finite differences, Lagrange interpolation, and least squares regression. Each method has its own advantages and limitations, and the choice of method may depend on the specific data and its characteristics.

Similar threads

  • Calculus and Beyond Homework Help
Replies
9
Views
982
  • Precalculus Mathematics Homework Help
Replies
19
Views
12K
  • Linear and Abstract Algebra
Replies
3
Views
1K
  • Engineering and Comp Sci Homework Help
Replies
2
Views
3K
Replies
19
Views
2K
Replies
3
Views
621
  • Linear and Abstract Algebra
Replies
2
Views
2K
Replies
1
Views
2K
  • Precalculus Mathematics Homework Help
Replies
4
Views
1K
Replies
3
Views
303
Back
Top