To determine whether a value is a root of the polynomial [tex]\( p(x) = x^4 - 9x^2 - 4x + 12 \)[/tex], we need to evaluate the polynomial at that value and check if the result is zero. A root of the polynomial is a value [tex]\( x \)[/tex] for which [tex]\( p(x) = 0 \)[/tex].
Let's evaluate the polynomial at the given values and see which evaluations yield zero.
1. Evaluate [tex]\( p(x) \)[/tex] at [tex]\( x = -2 \)[/tex]
[tex]\[
p(-2) = (-2)^4 - 9(-2)^2 - 4(-2) + 12 = \boxed{0}
\][/tex]
2. Evaluate [tex]\( p(x) \)[/tex] at [tex]\( x = 2 \)[/tex]
[tex]\[
p(2) = 2^4 - 9 \cdot 2^2 - 4 \cdot 2 + 12 = \boxed{-16}
\][/tex]
From these evaluations, we see that:
- [tex]\( p(-2) = 0 \)[/tex], indicating that [tex]\(-2\)[/tex] is a root of the polynomial.
- [tex]\( p(2) = -16 \)[/tex], indicating that [tex]\(2\)[/tex] is not a root of the polynomial.
Therefore, the value [tex]\(-2\)[/tex] is a root of [tex]\( p(x) \)[/tex].
To summarize, among the given options, [tex]\((\pm 2)\)[/tex] was initially considered:
- [tex]\(\pm 2\)[/tex] includes [tex]\(-2\)[/tex], which is a root of the polynomial.
- [tex]\(\pm 2\)[/tex] also includes [tex]\(2\)[/tex], which is not a root of the polynomial.
Thus, [tex]\(\pm 2\)[/tex] is partially correct. Specifically, only [tex]\(-2\)[/tex] is a root. The other provided values are not evaluated in this specific instruction, so no conclusion is drawn about them from this information alone.
