We need to find a polynomial of degree 4, $f(x)$, that passes through the points:

• (-3, -309)
• (-1, -5)
• (0, 3)
• (2, 1)
• (3, -117)

General form of a degree 4 polynomial:

$f(x) = ax^4 + bx^3 + cx^2 + dx + e$

Since the polynomial must pass through the five given points, we substitute each point $(x, y)$ into the general form to create a system of equations for the coefficients $a$, $b$, $c$, $d$, and $e$.

1. Substituting $(-3, -309)$: $-309 = a(-3)^4 + b(-3)^3 + c(-3)^2 + d(-3) + e$ $-309 = 81a - 27b + 9c - 3d + e$

2. Substituting $(-1, -5)$: $-5 = a(-1)^4 + b(-1)^3 + c(-1)^2 + d(-1) + e$ $-5 = a - b + c - d + e$

3. Substituting $(0, 3)$: $3 = a(0)^4 + b(0)^3 + c(0)^2 + d(0) + e$ $3 = e$

4. Substituting $(2, 1)$: $1 = a(2)^4 + b(2)^3 + c(2)^2 + d(2) + e$ $1 = 16a + 8b + 4c + 2d + e$

5. Substituting $(3, -117)$: $-117 = a(3)^4 + b(3)^3 + c(3)^2 + d(3) + e$ $-117 = 81a + 27b + 9c + 3d + e$

Now we can substitute $e = 3$ from the third equation into the other equations and solve the system of linear equations for $a$, $b$, $c$, and $d$.

Let me calculate that system.The solution to the system of equations gives the coefficients:

$a = -3, \quad b = 3, \quad c = 3, \quad d = 5, \quad e = 3$

Thus, the polynomial is:

$f(x) = -3x^4 + 3x^3 + 3x^2 + 5x + 3$

Would you like more details on how the system was solved, or do you have any other questions?

Here are 5 related questions to explore further:

1. How would you verify that this polynomial goes through the given points?
2. How does changing the degree of a polynomial affect its shape?
3. What happens if a polynomial of degree 4 only passes through 3 points? Is it still unique?
4. How does the behavior of the graph change near its roots?
5. Can you explain how polynomial coefficients affect the turning points of the graph?

Tip: When finding polynomials, the degree always corresponds to the number of roots (with multiplicity) or turns the graph can make.