- The Rational Root Theorem requires the numerator \(p\) to be a divisor of the constant term (\(10\)) and the denominator \(q\) to be a divisor of the leading coefficient (\(4\)).
- Tested values \(\pm \frac{1}{2}\), \(\pm \frac{5}{2}\), and \(\pm 2\) have numerators and denominators that divide \(10\) and \(4\) respectively.
- The value \(\pm \frac{2}{5}\) is excluded because its denominator \(5\) does not divide the leading coefficient \(4\).
- Final set of possible rational roots: \(\boxed{\pm \frac{1}{2}, \pm \frac{5}{2}, \pm 2}\).
Explanation
- Analyzing the Rational Root Theorem
The Rational Root Theorem states that for a polynomial equation \(a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0 = 0\) with integer coefficients, any rational root \(\frac{p}{q}\) (expressed in lowest terms) must satisfy two conditions: the numerator \(p\) is an integer divisor of the constant term \(a_0\), and the denominator \(q\) is an integer divisor of the leading coefficient \(a_n\).
Given the polynomial \(4x^3 + 9x^2 - x + 10 = 0\), we identify:
- Leading coefficient \(a_n = 4\)
- Constant term \(a_0 = 10\)
The integer divisors of the constant term \(a_0 = 10\) are \(p \in \{\pm 1, \pm 2, \pm 5, \pm 10\}\).
The integer divisors of the leading coefficient \(a_n = 4\) are \(q \in \{\pm 1, \pm 2, \pm 4\}\).
- Testing Potential Rational Roots
To determine if a rational value \(\frac{p}{q}\) is a potential root, we test if \(p\) is a factor of \(10\) and \(q\) is a factor of \(4\).
- For \(\pm \frac{1}{2}\): Here \(p=1\) (a factor of \(10\)) and \(q=2\) (a factor of \(4\)). This is a possible root.
- For \(\pm \frac{2}{5}\): Here \(p=2\) (a factor of \(10\)) but \(q=5\) (not a factor of \(4\)). This is NOT a possible root.
- For \(\pm \frac{5}{2}\): Here \(p=5\) (a factor of \(10\)) and \(q=2\) (a factor of \(4\)). This is a possible root.
- For \(\pm 2\): Here \(p=2\) (a factor of \(10\)) and \(q=1\) (a factor of \(4\)). This is a possible root.
- Conclusion
Based on the analysis of the divisors, the rational values that can potentially be roots of the polynomial \(4x^3 + 9x^2 - x + 10 = 0\) according to the Rational Root Theorem are:
- \(\pm \frac{1}{2}\)
- \(\pm \frac{5}{2}\)
- \(\pm 2\)
The value \(\pm \frac{2}{5}\) is not a potential root because \(5\) is not a divisor of the leading coefficient \(4\). Therefore, the correct values are \(\pm \frac{1}{2}\), \(\pm \frac{5}{2}\), and \(\pm 2\).
Examples
The Rational Root Theorem is a powerful tool in algebra for simplifying the search for roots of high-degree polynomials. For example, if an engineer is designing a structural component modeled by a cubic function, finding the precise roots helps identify critical equilibrium points or stress boundaries. By narrowing down the infinite set of real numbers to a finite set of rational possibilities, the theorem allows for efficient problem-solving in mechanical and civil engineering simulations.