Which numbers are potential roots of the function? Choose three correct answers.
$p(x)=x^4+22 x^2-16 x-12$
To identify the roots of the polynomial \(p(x) = x^4 + 22x^2 - 16x - 12\), we examine the potential rational roots using the Rational Root Theorem. According to this theorem, any rational root of a polynomial with integer coefficients, \(\frac{p}{q}\), must have \(p\) as a factor of the constant term \(-12\) and \(q\) as a factor of the leading coefficient \(1\). Therefore, the only possible rational roots are divisors of \(-12\), which are \(\{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12\}\).
We compare our list of potential rational roots with the candidates provided in the question: \(\{\pm 6, \pm 1, \pm \frac{1}{3}, \pm 8, \pm \frac{11}{2}, \pm 3\}\). Any candidate that is not a divisor of \(-12\) cannot be a rational root of the polynomial. This eliminates \(\pm \frac{1}{3}, \pm 8, \pm \frac{11}{2}\) immediately. We are left with \(\{\pm 6, \pm 1, \pm 3\}\) as the only candidates that could possibly be rational roots.
We test the remaining candidates \(\{6, -6, 1, -1, 3, -3\}\) by calculating \(p(x)\) for each:
None of the provided candidates are roots of the polynomial \(p(x)\). It appears that the polynomial \(p(x) = x^4 + 22x^2 - 16x - 12\) does not have any of the provided values as its roots.
Determining the roots of a polynomial is essential in various fields, such as engineering and physics, where roots represent equilibrium points or critical values in a system. For instance, in structural analysis, finding the roots of a characteristic equation helps determine the stability of a structure under load. If a specific set of test conditions does not produce a stable equilibrium (i.e., yields no roots), engineers know they must explore different parameters to ensure the safety and reliability of their designs. Understanding how to apply the Rational Root Theorem allows professionals to efficiently narrow down potential solutions.