Consider the following points.
(−2, 28), (−1, 0), (0, −6), (1, −8), (2, 0)
(a) Determine the polynomial function of least degree whose graph passes through the given points.
p(x) =

(b) Sketch the graph of the polynomial function, showing the given points.

### Part (a): Determine the Polynomial Function

To find the polynomial function of least degree that passes through the given points, we can assume a polynomial of the form:

\[ p(x) = ax^4 + bx^3 + cx^2 + dx + e \]

Since there are 5 points given, a polynomial of degree 4 is needed to uniquely determine the function. We will use the points to set up a system of equations to solve for the coefficients \( a \), \( b \), \( c \), \( d \), and \( e \).

1. For \( (-2, 28) \):

\[
p(-2) = 16a - 8b + 4c - 2d + e = 28
\]

2. For \( (-1, 0) \):

\[
p(-1) = a - b + c - d + e = 0
\]

3. For \( (0, -6) \):

\[
p(0) = e = -6
\]

4. For \( (1, -8) \):

\[
p(1) = a + b + c + d + e = -8
\]

5. For \( (2, 0) \):

\[
p(2) = 16a + 8b + 4c + 2d + e = 0
\]

Substituting \( e = -6 \) into the other equations:

1. \( 16a - 8b + 4c - 2d - 6 = 28 \) → \( 16a - 8b + 4c - 2d = 34 \)
2. \( a - b + c - d - 6 = 0 \) → \( a - b + c - d = 6 \)
3. \( a + b + c + d - 6 = -8 \) → \( a + b + c + d = -2 \)
4. \( 16a + 8b + 4c + 2d - 6 = 0 \) → \( 16a + 8b + 4c + 2d = 6 \)

This gives us a system of four equations:

\[
\begin{align*}
16a - 8b + 4c - 2d & = 34 \\
a - b + c - d & = 6 \\
a + b + c + d & = -2 \\
16a + 8b + 4c + 2d & = 6 \\
\end{align*}
\]

Now, we can solve this system to find the values of \( a \), \( b \), \( c \), and \( d \).

### Solving the System

Let's solve these equations step by step.

1. From equation \( e = -6 \), we've already found \( e \).
2. We can use elimination or substitution methods to find \( a \), \( b \), \( c \), and \( d \).

I'll solve these equations.The solution to the system of equations gives the coefficients:

\[
a = 1, \quad b = -1, \quad c = 1, \quad d = -3, \quad \text{and} \quad e = -6
\]

So, the polynomial function \( p(x) \) of least degree that

Learn how to find the polynomial function of least degree that passes through given points (-2, 28), (-1, 0), (0, -6), (1, -8), and (2, 0). This problem involves polynomial interpolation using a system of equations. Suitable for advanced high school students.

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