SOLUTION: Determine the function, f(x), in factored form, with integral coefficients, such that is has the following properties: 1) Third differences are equal to -18 2) The point (1/2, 0) i

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Question 1185986: Determine the function, f(x), in factored form, with integral coefficients, such that is has the following properties: 1) Third differences are equal to -18 2) The point (1/2, 0) is an x-intercept 3) The remainder is -60 when f(x) is divided by (x+2) 4) The y-intercept is -18. Show your work
Answer by greenestamps(13209) About Me  (Show Source):
You can put this solution on YOUR website!


A wonderful problem, requiring the use of a lot of different concepts to solve.

I especially like the constant third differences information; that is something that is missing from the curriculum in probably most high school math classes.

(1) The (constant) third differences are 18.

That means the polynomial is degree 3 (constant 3rd differences) with leading coefficient -3 (-18/3!). The polynomial function is

f%28x%29+=+%28-3%29%28x-a%29%28x-b%29%28x-c%29

(2) (1/2,0) is an x-intercept.

That means one of the linear factors is (x-1/2). The polynomial function is

f%28x%29+=+%28-3%29%28x-1%2F2%29%28x-b%29%28x-c%29

(3) The remainder is -60 when f(x) is divided by (x+2).

That means f(-2)=-60.

%28-3%29%28-5%2F2%29%28-2-b%29%28-2-c%29=-60
%2815%2F2%29%28b%2B2%29%28c%2B2%29=-60
bc%2B2b%2B2c%2B4=-8
bc%2B2b%2B2c=-12

(4) The y-intercept is -18.

That means f(0)=-18.

%28-3%29%28-1%2F2%29%28-b%29%28-c%29=-18
%283%2F2%29bc=-18
bc=-12

Combine the results from (3) and (4).

-12%2B2b%2B2c=-12
2b%2B2c=0
b=-c

Combine that result with (4).

%28-c%29%28c%29=-12
c%5E2=12
c=2%2Asqrt%283%29

b and c are 2sqrt%283%29 and -2sqrt%283%29

The function still in factored form is

f%28x%29+=+%28-3%29%28x-1%2F2%29%28x-2sqrt%283%29%29%28x%2B2sqrt%283%29%29

or, in a nicer form,

f%28x%29+=+%28-3%29%28x-1%2F2%29%28x%5E2-12%29

A graph, showing the x-intercept (1/2,0) and the y-intercept (0,-18):

graph%28400%2C400%2C-5%2C5%2C-50%2C50%2C%28-3%29%28x-1%2F2%29%28x%5E2-12%29%29