To find the potential rational roots of the polynomial function [tex]\( f(x) = 5x^5 + \frac{16}{5}x - 3 \)[/tex], we use the Rational Root Theorem. This theorem helps us determine the possible rational roots of a polynomial equation.
Here's a step-by-step explanation:
1. Identify the constant term and leading coefficient:
- The constant term is [tex]\( -3 \)[/tex].
- The leading coefficient is [tex]\( 5 \)[/tex].
2. List the factors of the constant term [tex]\( -3 \)[/tex]:
- The factors of [tex]\( -3 \)[/tex] are [tex]\( \pm 1 \)[/tex] and [tex]\( \pm 3 \)[/tex].
3. List the factors of the leading coefficient [tex]\( 5 \)[/tex]:
- The factors of [tex]\( 5 \)[/tex] are [tex]\( \pm 1 \)[/tex] and [tex]\( \pm 5 \)[/tex].
4. Form the potential rational roots:
- According to the Rational Root Theorem, any potential rational root, expressed as [tex]\( \frac{p}{q} \)[/tex], must have [tex]\( p \)[/tex] as a factor of the constant term and [tex]\( q \)[/tex] as a factor of the leading coefficient.
- Therefore, the potential rational roots are obtained by combining the factors of the constant term with the factors of the leading coefficient.
We get the following potential rational roots:
- [tex]\(\pm 1\)[/tex] (corresponding to [tex]\(\frac{\pm 1}{1}\)[/tex])
- [tex]\(\pm \frac{1}{5}\)[/tex] (corresponding to [tex]\(\frac{\pm 1}{5}\)[/tex])
- [tex]\(\pm 3\)[/tex] (corresponding to [tex]\(\frac{\pm 3}{1}\)[/tex])
- [tex]\(\pm \frac{3}{5}\)[/tex] (corresponding to [tex]\(\frac{\pm 3}{5}\)[/tex])
5. List all the potential rational roots:
- The potential rational roots derived are [tex]\( 1, -1, \frac{1}{5}, -\frac{1}{5}, 3, -3, \frac{3}{5}, -\frac{3}{5} \)[/tex].
Therefore, the potential rational roots of the polynomial function [tex]\( f(x) = 5x^5 + \frac{16}{5}x - 3 \)[/tex] are:
[tex]\[ 1, -1, \frac{1}{5}, -\frac{1}{5}, 3, -3, \frac{3}{5}, -\frac{3}{5} \][/tex]
