The polynomial f is divisible by g . Calculate p=f(0)*f(1)*...*f(2009) if f=x^3-4x^2+x+6 and g=x^2-x-2.

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We'll re-write g as a product of linear factors, using it's roots.

x1 = [1 + sqrt(1 + 8)]/2

x1 = (1+3)/2

x1 = 2

x2 = (1-3)/2

x2 = -1

Therefore, g = (x-2)(x+1)

Since f is divisible by g, it means that we can write f:

f = (x-2)(x+1)(x+a), where a is the 3rd root of f(x).

We'll compute a = 3

f = (x-2)(x+1)(x-3)

Now, we'll calculate the product:

P = f(0)*f(1)*...*f(2009)

Since 2 and 3 are the roots of f(x), it means that substituted in the expression of f(x), will cancel it.

f(2) = 0

f(3) = 0

P = f(0)*f(1)*0*0*...*f(2009) => P = 0

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