- The Rational Root Theorem states that any rational root \(\frac{p}{q}\) of a polynomial has \(p\) as a factor of the constant term and \(q\) as a factor of the leading coefficient.
- For \(q(x) = 6x^3 + 19x^2 - 15x - 28\), the factors of the constant \(-28\) are \(\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28\) and the factors of the leading coefficient \(6\) are \(\pm 1, \pm 2, \pm 3, \pm 6\).
- Testing the options, \(\pm \frac{2}{3}\), \(\pm 14\), and \(\pm \frac{7}{2}\) satisfy these conditions.
- The three correct choices are: \(\boxed{b, c, d}\).
Explanation
- Applying the Rational Root Theorem
The Rational Root Theorem provides a method to identify potential rational roots for a polynomial with integer coefficients. For a polynomial of the form \(P(x) = a_n x^n + \dots + a_0\), any rational root \(x = \frac{p}{q}\) (in simplest form) must satisfy: \(p\) is a factor of the constant term \(a_0\), and \(q\) is a factor of the leading coefficient \(a_n\).
For the given polynomial \(q(x) = 6x^3 + 19x^2 - 15x - 28\):
- The leading coefficient \(a_n\) is \(6\). The factors of \(6\) are \(\{\pm 1, \pm 2, \pm 3, \pm 6\}\). These are our possible values for \(q\).
- The constant term \(a_0\) is \(-28\). The factors of \(28\) are \(\{\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28\}\). These are our possible values for \(p\).
- Therefore, a rational number \(\frac{p}{q}\) is a potential root if \(p\) is a divisor of \(28\) and \(q\) is a divisor of \(6\).
- Evaluating Potential Roots
We evaluate each given option to see if it fits the required form \(\frac{p}{q}\) where \(p\) is a factor of \(28\) and \(q\) is a factor of \(6\):
- a. \(\pm \frac{1}{7}\): Here \(p=1\) (a factor of \(28\)) and \(q=7\). However, \(7\) is not a factor of \(6\). This is not a potential root.
- b. \(\pm \frac{2}{3}\): Here \(p=2\) (a factor of \(28\)) and \(q=3\) (a factor of \(6\)). This is a potential root.
- c. \(\pm 14\): This can be written as \(\frac{14}{1}\). Here \(p=14\) (a factor of \(28\)) and \(q=1\) (a factor of \(6\)). This is a potential root.
- d. \(\pm \frac{7}{2}\): Here \(p=7\) (a factor of \(28\)) and \(q=2\) (a factor of \(6\)). This is a potential root.
- e. \(\pm \frac{3}{5}\): Here \(p=3\) (not a factor of \(28\)) and \(q=5\) (not a factor of \(6\)). This is not a potential root.
- f. \(\pm 6\): This can be written as \(\frac{6}{1}\). While \(q=1\) is a factor of \(6\), \(p=6\) is not a factor of \(28\). This is not a potential root.
The three correct choices are b, c, and d.
Examples
The Rational Root Theorem is a powerful tool in algebra used to solve polynomial equations, which appear frequently in engineering and physics when modeling physical systems. For example, if you are designing a suspension bridge and need to find the specific tension points of a support cable modeled by a high-degree polynomial, applying this theorem helps you narrow down potential equilibrium positions. By systematically identifying possible roots, engineers can solve complex equations efficiently and ensure the stability and safety of their designs.