To determine which of the numbers are potential roots of the polynomial [tex]\( p(x) = x^3 + 6x^2 - 7x - 60 \)[/tex] using the Rational Root Theorem, we start by identifying the potential rational roots.
The Rational Root Theorem states that any rational root, expressed as a fraction [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term and [tex]\( q \)[/tex] is a factor of the leading coefficient, could be a potential root. In this polynomial's case:
- The constant term is [tex]\(-60\)[/tex], and its factors are [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60\)[/tex].
- The leading coefficient is 1, and its factors are [tex]\(\pm 1\)[/tex].
Therefore, the potential rational roots are all the factors of the constant term [tex]\((\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60\)[/tex]) divided by [tex]\(\pm 1\)[/tex], giving us [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 5, \pm 6, \pm 10, \pm 12, \pm 15, \pm 20, \pm 30, \pm 60\)[/tex] as potential rational roots.
Among the numbers given in the problem, we check:
- [tex]\(-10\)[/tex]: This is a potential root since it is on our list of potential rational roots.
- [tex]\(-7\)[/tex]: This is not a potential root because it is not a factor of [tex]\(-60\)[/tex].
- [tex]\(-5\)[/tex]: This is a potential root since it is on our list of potential rational roots.
- [tex]\(3\)[/tex]: This is a potential root since it is on our list of potential rational roots.
- [tex]\(15\)[/tex]: This is a potential root since it is on our list of potential rational roots.
- [tex]\(24\)[/tex]: This is not a potential root because it is not a factor of [tex]\(-60\)[/tex].
To verify whether these potential roots are actual roots, we evaluate the polynomial [tex]\( p(x) \)[/tex] at each potential root:
- For [tex]\(-10\)[/tex]: [tex]\( p(-10) \neq 0 \)[/tex].
- For [tex]\(-5\)[/tex]: [tex]\( p(-5) = 0 \)[/tex].
- For [tex]\(3\)[/tex]: [tex]\( p(3) = 0 \)[/tex].
- For [tex]\(15\)[/tex]: [tex]\( p(15) \neq 0 \)[/tex].
Thus, the numbers from the given list that are actual roots of the polynomial [tex]\( p(x) = x^3 + 6x^2 - 7x - 60 \)[/tex] are:
[tex]\[
-5 \quad \text{and} \quad 3
\][/tex]
