Exercise7
In a previous math class, you have probably seen the fact that, if we are given two points in the plane, then there is a unique line passing through both of them. In this problem, we will begin with the four points on the left below and ask to find a polynomial that passes through these four points as shown on the right.
A degree three polynomial can be written as
\begin{equation*} p(x) = a + bx + cx^2 + dx^3 \end{equation*}where \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\) are coefficients that we would like to determine. Since we want the polynomial to pass through the point \((3,1)\text{,}\) we should require that
\begin{equation*} p(3) = a + 3b + 9c + 27d = 1 \text{.} \end{equation*}In this way, we obtain a linear equation for the coefficients \(a\text{,}\) \(b\text{,}\) \(c\text{,}\) and \(d\text{.}\)
Write the four linear equations for the coefficients obtained by requiring that the graph of the polynomial \(p(x)\) passes through the four points above.
Write the augmented matrix corresponding to this system of equations and use the Sage cell below to solve for the coefficients.
Write the polynomial \(p(x)\) that you found and check your work by graphing it in the Sage cell below and verifying that it passes through the four points. To plot a function over a range, you may use a command like plot(1 + x 2x^2, xmin = 1, xmax = 4).

Rather than looking for a degree three polynomial, suppose we wanted to find a polynomial that passes through the four points and that has degree two, such as
\begin{equation*} p(x) = a + bx + cx^2 \text{.} \end{equation*}Solve the system of equations for the coefficients. What can you say about the existence and uniqueness of the solutions?

Rather than looking for a degree three polynomial, suppose we wanted to find a polynomial that passes through the four points and that has degree four, such as
\begin{equation*} p(x) = a + bx + cx^2 + dx^3 + ex^4 \text{.} \end{equation*}Solve the system of equations for the coefficients. What can you say about the existence and uniqueness of the solutions?
Suppose you had 10 points and you wanted to find a polynomial passing through each of them. What should the degree of the polynomial be to guarantee that there is exactly one such polynomial? Explain your response.