Question 555126: Find a third-degree polynomial function such that f(0) = 18 and whose zeros are -1, 2, and 3. Using complete sentences, explain how you found it.
Answer by htmentor(1343)   (Show Source):
You can put this solution on YOUR website! The formula for a 3rd degree polynomial is:
f(x) = ax^3 + bx^2 + cx + d
In order to find the polynomial, we need to find a, b, c and d
Given: f(0)=18, zeros -1, 2, 3
f(0) = 18 = d
The 3 zeros of the functions give us 3 equations in a, b, c:
f(-1) = 0 = a(-1)^3 + b(-1)^2 - c + 18
f(2) = 0 = a(2)^3 + b(2)^2 + 2c + 18
f(3) = 0 = a(3)^3 + b(3)^2 + 3c + 18
These give:
-a + b + c = -18
8a + 4b + 2c = -18
27a + 9b + 3c = -18
Solve this system of equations using your favorite method (substitution, etc.)
The solution is: a=3, b=-12, c=3
So the function is: f(x) = 3x^3 -12x^2 + 3x + 18
The graph is below.
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