Answer:
The polynomial equation that passes through the points is
Step-by-step explanation:
Suppose you have a function y = f(x) which goes through these points
A(-5,-3), B(-2,3). C(3,3), D(6,19)
there is a polynomial P(x) of degree 3 which goes through these point.
We use the fact that <em>four distinct points will determine a cubic function.</em>
P(x) is the degree 3 polynomial through the 4 points, a standard way to write it is
Next replace the given points one by one, which leads to a system of 4 equations and 4 variables (namely a,b,c,d)
We can rewrite this system as follows:
To use the Gaussian Elimination we need to express the system of linear equations in matrix form (<em>the matrix equation Ax=b</em>).
The coefficient matrix (A) for the above system is
the variable matrix (x) is
and the constant matrix (b) is
We also need the augmented matrix, it is obtained by appending the columns of the coefficient matrix and the constant matrix.
To transform the augmented matrix to the reduced row echelon form we need to follow these steps:
- Subtract row 1 from row 2
- Subtract row 1 from row 3
- Subtract row 1 from row 4
- Divide row 2 by 3
- Add row 2 multiplied by 5 to row 1
- Subtract row 2 multiplied by 8 from row 3
- Subtract row 2 multiplied by 11 from row 4
- Divide row 3 by 40
- Add row 3 multiplied by 10 to row 1
- Add row 3 multiplied by 7 to row 2
- Subtract row 3 multiplied by 88 from row 4
- Divide row 4 by 264
- Subtract row 4 multiplied by 30 from row 1
- Subtract row 4 multiplied by 11 from row 2
- Add row 4 multiplied by 4 to row 3
From the reduced row-echelon form the solutions are:
The polynomial P(x) is:
We can check our solution plotting the polynomial and checking that it passes through the points.
