DIRECTIONS: Complete the following problems to the best of your ability. Be sure

Question

DIRECTIONS: Complete the following problems to the best of your ability. Be sure to include all relevant work for your solution. Insufficient work may result in a deduction of points. 1) (4 points) Determine a polynomial equation that passes through the following points. In order to receive full credit, set up a system of equations to describe the situation. Set up an augmented matrix to describe the system. You may use your graphing calculator or other technology to obtain reduced row echelon form. 4 52 -2 18

Explanation / Answer

there are 4 points given

hence we can fit (n-1) = 3 degree polynomial

y = a + bx + cx^2 +dx^3

X =

     1     1     1     1
     1     2     4     8
     1     3     9    27
     1     4    16    64

y =

    -2
     2
    18
    52

Xb = y

b = inv (X'X) * X'y

First, we must convert the matrix to reduced row echelon form:

Add (-1 * row1) to row2

Add (-1 * row1) to row3

Add (-1 * row1) to row4

Add (-2 * row2) to row3

Add (-3 * row2) to row4

Divide row3 by 2

Add (-6 * row3) to row4

Divide row4 by 6

Add (-6 * row4) to row3

Add (-7 * row4) to row2

Add (-1 * row4) to row1

Add (-3 * row3) to row2

Add (-1 * row3) to row1

Add (-1 * row2) to row1

Because we have only performed linear operations on rows, the non-zero rows in the reduced row echelon form of the matrix comprise a Basis for the Row Space of the matrix.
(Note that this is not true of the Column Space; the Column Space certainly changes as you perform row operations.)
The rows highlighted below in yellow comprise a Basis for the Row Space of our matrix:

hence

d = 1 ,c = 0 , b = -3 ,a = 0

hence

cubic polynmial is x^3 - 3x

Please rate

1 1 1 1 -2 0 1 3 7 4 1 3 9 27 18 1 4 16 64 52

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