Given the polynomial function
f(x)=x^3-4x^2+x+6
1) name all of the possible roots for the function using the rational root theorem.

2) prove 2 is one of the zeroes of this function (show or describe your process).

3) name the three actual roots of this function and describe how you found them.

4) write the original polynomial function in factored form and describe how you found those factors.

Explanation:

1) The possible roots are ±1, ±2, ±3, ±6. These are the factors of the trailing constant (6) divided by the factors of the leading coefficient (1). When the leading coefficient is 1, the possible roots are the factors of the constant.

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2) The polynomial is easier to evaluate when it is written in Horner form:

  f(x) = ((x -4)x +1)x +6

To show that 2 is a zero, we want to find f(2):

  f(2) = ((2 -4)2 +1)2 +6 = (-4 +1)2 +6 = -6 +6 = 0

  f(2) = 0, so 2 is one of the zeros of this function

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3) Using synthetic division (attached) or polynomial long division, we can divide the given polynomial by (x-2) to find the remaining factors. This division gives (x^2 -2x -3), which can be factored as (x -3)(x +1), so the three actual roots are ...

  x = 2 (from above), x = 3, x = -1 (from our factorization)

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4) In factored form, the polynomial can be written ...

  f(x) = (x +1)(x -2)(x -3)

The first factor was found from the fact that 2 was given as a zero of the function. For any zero "a", a factor of the polynomial is (x-a).

The remaining factors were found by factoring the quadratic trinomial that resulted from the division of f(x) by x-2. That trinomial is x^2 -2x -3.

There are a number of methods that can be used to factor x^2 -2x -3. Again, the rational root theorem can help. It suggests that ±1 and ±3 are possible roots.

We want to choose two of these that have a sum of 2 (the opposite of the x-coefficient). Those would be 3 and -1. If 3 and -1 are roots, then the remaining two factors are (x-3) and (x+1).

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Alternatively, the roots can be found using a graphing calculator, and those root values made into factors as above. Since the leading coefficient of f(x) is 1, all of the binomial factors will have leading coefficients of 1. The roots are shown to be -1, 2, 3, so the factors are (x+1)(x-2)(x-3).