To find the set of possible rational zeros for the polynomial function [tex]\( f(x) = 5x^3 - 8x^2 + 3 \)[/tex], we can apply the Rational Root Theorem. This theorem states that any rational root, written in the form [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.
1. Identify the constant term and the leading coefficient:
- The constant term (the term without [tex]\(x\)[/tex]) in [tex]\( f(x) = 5x^3 - 8x^2 + 3 \)[/tex] is 3.
- The leading coefficient (the coefficient of the highest power of [tex]\(x\)[/tex]) is 5.
2. Find the factors of the constant term (3):
- The factors of 3 are [tex]\(\pm 1, \pm 3\)[/tex].
3. Find the factors of the leading coefficient (5):
- The factors of 5 are [tex]\(\pm 1, \pm 5\)[/tex].
4. Form all possible fractions [tex]\(\frac{p}{q}\)[/tex] where [tex]\(p\)[/tex] is a factor of the constant term and [tex]\(q\)[/tex] is a factor of the leading coefficient:
- List of [tex]\(\frac{p}{q}\)[/tex] combinations:
[tex]\[
\frac{\pm 1}{\pm 1}, \frac{\pm 1}{\pm 5}, \frac{\pm 3}{\pm 1}, \frac{\pm 3}{\pm 5}
\][/tex]
5. Simplify the fractions to obtain all unique values:
[tex]\[
\begin{align*}
\frac{1}{1} &= 1 \\
\frac{-1}{1} &= -1 \\
\frac{1}{-1} &= 1 \\
\frac{-1}{-1} &= -1 \\
\frac{1}{5} &= 0.2 \\
\frac{-1}{5} &= -0.2 \\
\frac{1}{-5} &= -0.2 \\
\frac{-1}{-5} &= 0.2 \\
\frac{3}{1} &= 3 \\
\frac{-3}{1} &= -3 \\
\frac{3}{-1} &= -3 \\
\frac{-3}{-1} &= 3 \\
\frac{3}{5} &= 0.6 \\
\frac{-3}{5} &= -0.6 \\
\frac{3}{-5} &= -0.6 \\
\frac{-3}{-5} &= 0.6 \\
\end{align*}
\][/tex]
Combining all unique values, we get the set of possible rational zeros:
[tex]\[
\{-3, -1, -\frac{3}{5}, -\frac{1}{5}, 0.2, 0.6, \frac{1}{5}, \frac{3}{5}, 3, 1\}
\][/tex]
However, typically in simpler forms for rational expressions, we combine and remove duplicates, ensuring clarity:
[tex]\[
\{-3, -1, -0.6, -0.2, 0.2, 0.6, 1, 3 \} = \left\{-3, -1, -\frac{3}{5}, -\frac{1}{5}, \frac{1}{5}, \frac{3}{5}, 1, 3\right\}
\][/tex]
Therefore, the set of possible rational zeros is:
[tex]\[
\{-3, -1, -\frac{3}{5}, -\frac{1}{5}, \frac{1}{5}, \frac{3}{5}, 1, 3\}
\][/tex]
