5th Degree Polynomial Matrix

In summary, the given equation y=a0+a1x+a2x^2+a3x^3+a4x^4+a5x^5 (where a0, a1, ..., a5 are fixed real numbers) passes through the points (-2,21), (-1,7), (0,-10), (1,-8), (2,20), (3,9). The task is to write down a 6x6 matrix A such that the equation A(x,y)^T = y is satisfied, where x and y are the given points. This can be solved using basic linear algebra methods, such as setting up a system of equations with the unknown coefficients as variables.
  • #1
Bob Ho
18
0

Homework Statement


y=a0+a1x+a2x^2+a3x^3+a4x^4+a5x^5
(a0,a1...,a5, are fixed real numbers) passes through the points

(-2,21), (-1,7), (0,-10), (1,-8), (2,20), (3,9)

Question: Write down a 6x6 matrix A such that;


... .a0... ...21
. . .a1... ...7
. . .a2... ...-10
A . ..a3.. .= ...-8
. . .a4... ...20
. .a5... ...9




The Attempt at a Solution



I am unsure how to start this question. Do i substitute the points?

Any assistance would be obliged.
 
Last edited:
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  • #2
You need only very slight linear algebra knowledge to begin.

Your coefficients are the unknowns, and the x and y values are given as the points.
You will have six equations and six unknowns. The column of y values should be the rightmost column in your matrix. Just elementary row operations (if this is not too simple-minded) could take much time. Might you be allowed to use a software?
 

What is a 5th Degree Polynomial Matrix?

A 5th degree polynomial matrix is a mathematical object that contains polynomial expressions of degree 5 or less arranged in a rectangular array. It can be represented as a matrix, with rows and columns containing the coefficients of the polynomial terms.

What is the use of a 5th Degree Polynomial Matrix?

A 5th degree polynomial matrix is used in various fields of science and engineering, such as signal processing, control theory, and computer graphics. It can also be used to solve systems of polynomial equations and represent complex data structures.

How is a 5th Degree Polynomial Matrix constructed?

A 5th degree polynomial matrix is constructed by arranging the coefficients of the polynomial terms in a rectangular array, with each row representing a different polynomial. The number of columns in the matrix is equal to the degree of the highest polynomial in the matrix.

What are the operations that can be performed on a 5th Degree Polynomial Matrix?

The basic operations that can be performed on a 5th degree polynomial matrix include addition, subtraction, and multiplication. It can also be used in matrix transformations, such as rotation, scaling, and translation.

What are the advantages of using a 5th Degree Polynomial Matrix over other representations?

A 5th degree polynomial matrix provides a compact and efficient representation of polynomial equations, making it easier to perform computations and solve problems. It also allows for easy manipulation of the polynomial terms and can be easily extended to higher degrees.

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