Answer :
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.
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.
