Consider the function f(x)=x^3+6x^2+2x+12
sketched above.
The graph tells us something about the zeros of this funcion. Find the exact value all three zeros. "Exact value" means decimal approximations are not allowed.

To find the exact values of the zeros of the function \( f(x) = x^3 + 6x^2 + 2x + 12 \), we need to solve the equation \( f(x) = 0 \). Let's go through the process of factoring the cubic polynomial.

1. **Identify potential rational roots** using the Rational Root Theorem, which states that any rational solution, expressed as \(\frac{p}{q}\), must have \(p\) as a factor of the constant term (12) and \(q\) as a factor of the leading coefficient (1).

   Factors of 12 (constant term): \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)

   Factors of 1 (leading coefficient): \( \pm 1 \)

   Possible rational roots: \( \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12 \)

2. **Test the potential roots** by substituting them into the function and checking if they yield zero.

   - For \( x = -2 \):
     \[
     f(-2) = (-2)^3 + 6(-2)^2 + 2(-2) + 12 = -8 + 24 - 4 + 12 = 24
     \]
     \( f(-2) \neq 0 \)

   - For \( x = -3 \):
     \[
     f(-3) = (-3)^3 + 6(-3)^2 + 2(-3) + 12 = -27 + 54 - 6 + 12 = 33
     \]
     \( f(-3) \neq 0 \)

   - For \( x = -4 \):
     \[
     f(-4) = (-4)^3 + 6(-4)^2 + 2(-4) + 12 = -64 + 96 - 8 + 12 = 36
     \]
     \( f(-4) \neq 0 \)

   - For \( x = -1 \):
     \[
     f(-1) = (-1)^3 + 6(-1)^2 + 2(-1) + 12 = -1 + 6 - 2 + 12 = 15
     \]
     \( f(-1) \neq 0 \)

   - For \( x = -6 \):
     \[
     f(-6) = (-6)^3 + 6(-6)^2 + 2(-6) + 12 = -216 + 216 - 12 + 12 = 0
     \]
     \( f(-6) = 0 \)

     We have found a root, \( x = -6 \). Now, we use this root to factor the cubic polynomial.

3. **Factor the cubic polynomial** using synthetic division or long division.

   - We divide \( f(x) \) by \( (x + 6) \):

   Using synthetic division, we divide \( x^3 + 6x^2 + 2x + 12 \) by \( x + 6 \) and obtain \( x^2 - 4 \).

   \[
   x^3 + 6x^2 + 2x + 12 = (x + 6)(x^2 - 4)
   \]

4. **Solve the quadratic equation** \( x^2 - 4 = 0 \).

   \[
   x^2 - 4 = 0
   \]

   \[
   (x - 2)(x + 2) = 0
   \]

   Therefore, the solutions are \( x = 2 \) and \( x = -2 \).

The exact values of the zeros are \( \boxed{-6, -2, 2} \).

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Would you like further details or have any questions? Here are some related questions to consider:

1. How do you verify if a given value is a zero of a function?
2. What is the significance of finding the exact roots of a polynomial?
3. How can you determine if a polynomial has any complex roots?
4. What does the Rational Root Theorem tell us?
5. What is synthetic division, and how is it used in factoring polynomials?
6. How does the Fundamental Theorem of Algebra relate to the zeros of a polynomial?
7. What are the possible end behaviors of cubic functions?
8. How can graphing a function help identify its zeros?

**Tip:** When solving for roots of polynomials, always check if the polynomial can be factored easily before resorting to more complex methods like synthetic division or the quadratic formula.

Discover how to find the exact zeros of the cubic function x^3 + 6x^2 + 2x + 12 using the Rational Root Theorem and polynomial factoring techniques. This problem involves identifying rational roots and using synthetic division to factorize the polynomial. Suitable for advanced high school students.

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