SOLUTION: Determine the cubic function f(x) in factored form with integral coefficients, such that it has the following properties,
1) the third differences are equal to 18
2) the remainde
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-> SOLUTION: Determine the cubic function f(x) in factored form with integral coefficients, such that it has the following properties,
1) the third differences are equal to 18
2) the remainde
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Question 1186359: Determine the cubic function f(x) in factored form with integral coefficients, such that it has the following properties,
1) the third differences are equal to 18
2) the remainder f(x)/(x-2) is 27
3) x+1 is a factor
4) (1,6) is a point on f(x) Answer by greenestamps(13208) (Show Source):
(1) For a cubic polynomial, the third differences are constant, and that constant third difference is 3!=6 times the leading coefficient. With a constant difference of 18, the leading coefficient of f(x) is 3.
(2) If the remainder when f(x) is divided by (x-2) is 27, then f(2)=27.
(3) We are given that one of the factors is (x+1).
(4) (1,6) is a point on the graph of f(x) means f(1)=6.
From the given information, we know f(x) has the following factored (possibly only partially factored) form:
[1]
[2]
Solving [1] and [2] together finds b = -1 and c = 1.
So the quadratic factor is , which does not factor over the integers. So
