Answered

Using the Rational Root Theorem, select all the possible rational roots of the function.



$f(x)=3 x^3+x^2-38 x+24$


  • $\frac{2}{3}$
  • $-\frac{1}{4}$
  • $-\frac{1}{3}$
  • $4$
  • $-8$
  • $\frac{1}{2}$

Verified Answer :

  • Identify the leading coefficient \(a_n=3\) and constant term \(a_0=24\).
  • Find factors of \(a_0\) as \(\{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\}\) and factors of \(a_n\) as \(\{1, 3\}\).
  • Potential roots are formed by \(\pm \frac{p}{q}\) where \(p\) is a factor of 24 and \(q\) is a factor of 3.
  • Among the options, the possible rational roots are \(\boxed{\frac{2}{3}, -\frac{1}{3}, 4, -8}\).

Explanation

  1. Understanding the Rational Root Theorem

To identify the possible rational roots of the polynomial \(f(x) = 3x^3 + x^2 - 38x + 24\), we use the Rational Root Theorem. This theorem states that for a polynomial \(f(x) = a_n x^n + \dots + a_1 x + a_0\) with integer coefficients, any rational root expressed in lowest terms as \(\frac{p}{q}\) must satisfy the condition that \(p\) is a divisor of the constant term \(a_0\) and \(q\) is a divisor of the leading coefficient \(a_n\).

In our polynomial, the leading coefficient is \(a_n = 3\) and the constant term is \(a_0 = 24\).

  1. Determining Potential Rational Roots

First, list the divisors of the constant term \(a_0 = 24\). These are the potential values for the numerator \(p\):
\(p \in \{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\}\).

Next, list the positive divisors of the leading coefficient \(a_n = 3\). These are the potential values for the denominator \(q\):
\(q \in \{1, 3\}\).

Now, construct the set of all possible rational roots \(\frac{p}{q}\) by taking combinations of \(p\) and \(q\):

  • For \(q = 1\): \(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\)
  • For \(q = 3\): \(\pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{3}{3} (\pm 1), \pm \frac{4}{3}, \pm \frac{6}{3} (\pm 2), \pm \frac{8}{3}, \pm \frac{12}{3} (\pm 4), \pm \frac{24}{3} (\pm 8)\)

Combining these, the set of potential rational roots is:
\(\{\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24, \pm \frac{1}{3}, \pm \frac{2}{3}, \pm \frac{4}{3}, \pm \frac{8}{3}\}\).

  1. Testing the Given Options

We now test each given option against the identified set of potential rational roots:

  • \(\frac{2}{3}\): Included in the set.
  • \(-\frac{1}{4}\): Not included (4 is not a divisor of 3).
  • \(-\frac{1}{3}\): Included in the set.
  • \(4\): Included in the set.
  • \(-8\): Included in the set.
  • \(\frac{1}{2}\): Not included (2 is not a divisor of 3).

The possible rational roots among the provided options are \(\frac{2}{3}\), \(-\frac{1}{3}\), \(4\), and \(-8\).

Examples

The Rational Root Theorem is a fundamental tool for solving polynomial equations encountered in various engineering and physical science problems. For instance, when modeling the trajectory of a projectile, the motion might be described by a cubic polynomial. Finding the roots of this polynomial helps determine the points in time or space when the object is at a specific height. By systematically identifying rational roots, we can simplify the polynomial and make the analysis of complex systems much more manageable.

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