- Identify a root as any value \(c\) that makes the polynomial \(p(c) = 0\).
- Observe that \(p(-2) = 0\), which meets the definition of a root.
- Observe that \(p(2) = -16\), which does not equal 0, so 2 is not a root.
- Conclude that the only root among the given values is \(-2\), so \(\boxed{-2}\).
Explanation
- Understanding Roots of Polynomials
The objective is to determine which of the values \(-2\) or \(2\) are roots of the polynomial \(p(x) = x^4 - 9x^2 - 4x + 12\). A root of a polynomial \(p(x)\) is defined as any value \(c\) such that \(p(c) = 0\). We are given the evaluations \(p(-2) = 0\) and \(p(2) = -16\). By comparing these values to the definition, we can identify the roots.
- Checking \(x = -2\)
We test whether \(x = -2\) is a root by checking if \(p(-2) = 0\). The problem statement directly provides that \(p(-2) = 0\). Since this satisfies the definition of a root, we conclude that \(-2\) is indeed a root of the polynomial \(p(x)\).
- Checking \(x = 2\)
Next, we test whether \(x = 2\) is a root by checking if \(p(2) = 0\). The problem statement provides that \(p(2) = -16\). Because \(p(2)
eq 0\), the value \(2\) does not satisfy the definition of a root. Therefore, \(2\) is not a root of the polynomial \(p(x)\).
- Conclusion
Based on our analysis, only the value \(-2\) satisfies the definition of a root. Thus, \(-2\) is the root of the given polynomial. The final answer is \(\boxed{-2}\).
Examples
Identifying roots is fundamental in solving polynomial equations. For instance, if you are modeling the trajectory of a ball, the roots of the position equation can tell you the exact moments in time when the ball is at ground level. Understanding that a root occurs when the output of a function is zero allows engineers and scientists to solve for critical points in physical systems or optimization problems.