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
![\left[\begin{array}{cccc}1&-5&25&-125\\1&-2&4&-8\\1&3&9&27\\1&6&36&216\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7D1%26-5%2625%26-125%5C%5C1%26-2%264%26-8%5C%5C1%263%269%2627%5C%5C1%266%2636%26216%5Cend%7Barray%7D%5Cright%5D)
the variable matrix (x) is
![\left[\begin{array}{c}a&b&c&d\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Da%26b%26c%26d%5Cend%7Barray%7D%5Cright%5D)
and the constant matrix (b) is
![\left[\begin{array}{c}-3&3&3&19\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7D-3%263%263%2619%5Cend%7Barray%7D%5Cright%5D)
We also need the augmented matrix, it is obtained by appending the columns of the coefficient matrix and the constant matrix.
![\left[\begin{array}{cccc|c}1&-5&25&-125&-3\\1&-2&4&-8&3\\1&3&9&27&3\\1&6&36&216&19\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%26-5%2625%26-125%26-3%5C%5C1%26-2%264%26-8%263%5C%5C1%263%269%2627%263%5C%5C1%266%2636%26216%2619%5Cend%7Barray%7D%5Cright%5D)
To transform the augmented matrix to the reduced row echelon form we need to follow these steps:
- Subtract row 1 from row 2

![\left[\begin{array}{cccc|c}1&-5&25&-125&-3\\0&3&-21&117&6\\1&3&9&27&3\\1&6&36&216&19\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%26-5%2625%26-125%26-3%5C%5C0%263%26-21%26117%266%5C%5C1%263%269%2627%263%5C%5C1%266%2636%26216%2619%5Cend%7Barray%7D%5Cright%5D)
- Subtract row 1 from row 3

![\left[\begin{array}{cccc|c}1&-5&25&-125&-3\\0&3&-21&117&6\\0&8&-16&152&6\\1&6&36&216&19\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%26-5%2625%26-125%26-3%5C%5C0%263%26-21%26117%266%5C%5C0%268%26-16%26152%266%5C%5C1%266%2636%26216%2619%5Cend%7Barray%7D%5Cright%5D)
- Subtract row 1 from row 4

![\left[\begin{array}{cccc|c}1&-5&25&-125&-3\\0&3&-21&117&6\\0&8&-16&152&6\\0&11&11&341&22\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%26-5%2625%26-125%26-3%5C%5C0%263%26-21%26117%266%5C%5C0%268%26-16%26152%266%5C%5C0%2611%2611%26341%2622%5Cend%7Barray%7D%5Cright%5D)
- Divide row 2 by 3

![\left[\begin{array}{cccc|c}1&-5&25&-125&-3\\0&1&-7&39&2\\0&8&-16&152&6\\0&11&11&341&22\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%26-5%2625%26-125%26-3%5C%5C0%261%26-7%2639%262%5C%5C0%268%26-16%26152%266%5C%5C0%2611%2611%26341%2622%5Cend%7Barray%7D%5Cright%5D)
- Add row 2 multiplied by 5 to row 1

![\left[\begin{array}{cccc|c}1&0&-10&-70&7\\0&1&-7&39&2\\0&8&-16&152&6\\0&11&11&341&22\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%26-10%26-70%267%5C%5C0%261%26-7%2639%262%5C%5C0%268%26-16%26152%266%5C%5C0%2611%2611%26341%2622%5Cend%7Barray%7D%5Cright%5D)
- Subtract row 2 multiplied by 8 from row 3

![\left[\begin{array}{cccc|c}1&0&-10&-70&7\\0&1&-7&39&2\\0&0&40&-160&-10\\0&11&11&341&22\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%26-10%26-70%267%5C%5C0%261%26-7%2639%262%5C%5C0%260%2640%26-160%26-10%5C%5C0%2611%2611%26341%2622%5Cend%7Barray%7D%5Cright%5D)
- Subtract row 2 multiplied by 11 from row 4

![\left[\begin{array}{cccc|c}1&0&-10&-70&7\\0&1&-7&39&2\\0&0&40&-160&-10\\0&0&88&-88&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%26-10%26-70%267%5C%5C0%261%26-7%2639%262%5C%5C0%260%2640%26-160%26-10%5C%5C0%260%2688%26-88%260%5Cend%7Barray%7D%5Cright%5D)
- Divide row 3 by 40

![\left[\begin{array}{cccc|c}1&0&-10&-70&7\\0&1&-7&39&2\\0&0&1&-4&-1/4\\0&0&88&-88&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%26-10%26-70%267%5C%5C0%261%26-7%2639%262%5C%5C0%260%261%26-4%26-1%2F4%5C%5C0%260%2688%26-88%260%5Cend%7Barray%7D%5Cright%5D)
- Add row 3 multiplied by 10 to row 1

![\left[\begin{array}{cccc|c}1&0&0&30&9/2\\0&1&-7&39&2\\0&0&1&-4&-1/4\\0&0&88&-88&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%260%2630%269%2F2%5C%5C0%261%26-7%2639%262%5C%5C0%260%261%26-4%26-1%2F4%5C%5C0%260%2688%26-88%260%5Cend%7Barray%7D%5Cright%5D)
- Add row 3 multiplied by 7 to row 2

![\left[\begin{array}{cccc|c}1&0&0&30&9/2\\0&1&0&11&1/4\\0&0&1&-4&-1/4\\0&0&88&-88&0\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%260%2630%269%2F2%5C%5C0%261%260%2611%261%2F4%5C%5C0%260%261%26-4%26-1%2F4%5C%5C0%260%2688%26-88%260%5Cend%7Barray%7D%5Cright%5D)
- Subtract row 3 multiplied by 88 from row 4

![\left[\begin{array}{cccc|c}1&0&0&30&9/2\\0&1&0&11&1/4\\0&0&1&-4&-1/4\\0&0&0&264&22\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%260%2630%269%2F2%5C%5C0%261%260%2611%261%2F4%5C%5C0%260%261%26-4%26-1%2F4%5C%5C0%260%260%26264%2622%5Cend%7Barray%7D%5Cright%5D)
- Divide row 4 by 264

![\left[\begin{array}{cccc|c}1&0&0&30&9/2\\0&1&0&11&1/4\\0&0&1&-4&-1/4\\0&0&0&1&1/12\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%260%2630%269%2F2%5C%5C0%261%260%2611%261%2F4%5C%5C0%260%261%26-4%26-1%2F4%5C%5C0%260%260%261%261%2F12%5Cend%7Barray%7D%5Cright%5D)
- Subtract row 4 multiplied by 30 from row 1

![\left[\begin{array}{cccc|c}1&0&0&0&2\\0&1&0&11&1/4\\0&0&1&-4&-1/4\\0&0&0&1&1/12\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%260%260%262%5C%5C0%261%260%2611%261%2F4%5C%5C0%260%261%26-4%26-1%2F4%5C%5C0%260%260%261%261%2F12%5Cend%7Barray%7D%5Cright%5D)
- Subtract row 4 multiplied by 11 from row 2

![\left[\begin{array}{cccc|c}1&0&0&0&2\\0&1&0&0&-2/3\\0&0&1&-4&-1/4\\0&0&0&1&1/12\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%260%260%262%5C%5C0%261%260%260%26-2%2F3%5C%5C0%260%261%26-4%26-1%2F4%5C%5C0%260%260%261%261%2F12%5Cend%7Barray%7D%5Cright%5D)
- Add row 4 multiplied by 4 to row 3

![\left[\begin{array}{cccc|c}1&0&0&0&2\\0&1&0&0&-2/3\\0&0&1&0&1/12\\0&0&0&1&1/12\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bcccc%7Cc%7D1%260%260%260%262%5C%5C0%261%260%260%26-2%2F3%5C%5C0%260%261%260%261%2F12%5C%5C0%260%260%261%261%2F12%5Cend%7Barray%7D%5Cright%5D)
From the reduced row-echelon form the solutions are:
![\left[\begin{array}{c}a=2&b=-2/3&c=1/12&d=1/12\end{array}\right]](https://tex.z-dn.net/?f=%5Cleft%5B%5Cbegin%7Barray%7D%7Bc%7Da%3D2%26b%3D-2%2F3%26c%3D1%2F12%26d%3D1%2F12%5Cend%7Barray%7D%5Cright%5D)
The polynomial P(x) is:

We can check our solution plotting the polynomial and checking that it passes through the points.