To determine the set of possible rational roots of the polynomial [tex]\( P(x) = x^3 + 5x^2 - 8x - 20 \)[/tex], we will use the Rational Root Theorem. This theorem states that any rational root, expressed in the form [tex]\(\frac{p}{q}\)[/tex], must have [tex]\(p\)[/tex] as a factor of the constant term and [tex]\(q\)[/tex] as a factor of the leading coefficient.
Here, the polynomial is [tex]\(P(x) = x^3 + 5x^2 - 8x - 20\)[/tex].
1. Identify the factors of the constant term (which is [tex]\(-20\)[/tex]):
- Factors of [tex]\(-20\)[/tex]: [tex]\(\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\)[/tex].
2. Identify the factors of the leading coefficient (which is [tex]\(1\)[/tex]):
- Factors of [tex]\(1\)[/tex]: [tex]\(\pm 1\)[/tex].
3. Use the Rational Root Theorem to list all possible rational roots [tex]\(\frac{p}{q}\)[/tex]:
- Since the denominators [tex]\(q\)[/tex] are all factors of 1, the rational roots must be simply the factors of the constant term [tex]\(\pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20\)[/tex].
Putting all this together, the possible rational roots are:
[tex]\[
\{ \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \}
\][/tex]
Therefore, the correct answer is:
C. [tex]\(\{ \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \}\)[/tex].
