To determine the real zeros of the polynomial function
[tex]\[ f(x) = x^3 + 2x^2 - 9x - 18 \][/tex]
we need to find the values of [tex]\( x \)[/tex] such that [tex]\( f(x) = 0 \)[/tex]. Here's a step-by-step process to find these real zeros:
1. Setting the Polynomial to Zero:
[tex]\[ x^3 + 2x^2 - 9x - 18 = 0 \][/tex]
2. Solution Analysis:
The result from solving the equation [tex]\( x^3 + 2x^2 - 9x - 18 = 0 \)[/tex] are the values of [tex]\( x \)[/tex] that satisfy the equation. These values are the real zeros of the polynomial.
3. Given Real Zeros:
From our calculations, the real zeros of the polynomial are:
[tex]\[ x = -3, x = -2, \text{ and } x = 3 \][/tex]
4. Verification of Choices:
Now we compare these real zeros with the choices provided:
- A. [tex]\(1,\, 2,\, 3\)[/tex]
- B. [tex]\(3,\, -3,\, 2\)[/tex]
- C. [tex]\(2,\, 3,\, -6\)[/tex]
- D. [tex]\(1,\, -1,\, 18\)[/tex]
- E. [tex]\(-2,\, 3,\, -3\)[/tex]
5. Comparison Results:
- Choice A: does not contain [tex]\(-3\)[/tex] or [tex]\(-2\)[/tex].
- Choice B: contains [tex]\(3\)[/tex] and [tex]\(-3\)[/tex] but not [tex]\(-2\)[/tex].
- Choice C: contains [tex]\(3\)[/tex] but does not contain [tex]\(-3\)[/tex] or [tex]\(-2\)[/tex].
- Choice D: does not contain [tex]\(-3\)[/tex], [tex]\(-2\)[/tex], or [tex]\(3\)[/tex].
- Choice E: contains all the correct numbers [tex]\(-2, 3,\)[/tex] and [tex]\(-3\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{-2,\, 3,\, -3} \][/tex]
Hence, the real zeros of the given polynomial function are provided in option E.
