Find the polynomial of degree 4 whose graph goes through the points 
(-3,-309), (-1,-5), (0,3), (2,1), (3, -117)

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.

Find the polynomial of degree 4 whose graph goes through the points (-3,-309), (-1,-5), (0,3), (2,1), (3,-117)

Find the degree 4 polynomial that passes through the points (-3,-309), (-1,-5), (0,3), (2,1), and (3,-117). Learn how to substitute these points into the general polynomial form to create and solve a system of equations. Ideal for high school students studying advanced algebra.

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