Math Problem Statement
Find the polynomial of degree 4 whose graph goes through the points (-3,-309), (-1,-5), (0,3), (2,1), (3, -117)
Solution
We need to find a polynomial of degree 4, , that passes through the points:
- (-3, -309)
- (-1, -5)
- (0, 3)
- (2, 1)
- (3, -117)
General form of a degree 4 polynomial:
Since the polynomial must pass through the five given points, we substitute each point into the general form to create a system of equations for the coefficients , , , , and .
-
Substituting :
-
Substituting :
-
Substituting :
-
Substituting :
-
Substituting :
Now we can substitute from the third equation into the other equations and solve the system of linear equations for , , , and .
Let me calculate that system.The solution to the system of equations gives the coefficients:
Thus, the polynomial is:
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:
- How would you verify that this polynomial goes through the given points?
- How does changing the degree of a polynomial affect its shape?
- What happens if a polynomial of degree 4 only passes through 3 points? Is it still unique?
- How does the behavior of the graph change near its roots?
- 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.
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Math Problem Analysis
Mathematical Concepts
Algebra
Polynomial Equations
Systems of Equations
Formulas
General form of a degree 4 polynomial: f(x) = ax^4 + bx^3 + cx^2 + dx + e
Substitute points into polynomial to form a system of linear equations
Theorems
Linear Equation Solving
Polynomial Interpolation
Suitable Grade Level
Grades 10-12