To answer this question, we will use the Rational Zero Theorem. This theorem states that any rational solution of the polynomial equation [tex]\( P(x) = 0 \)[/tex] is a fraction [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.
Given the polynomial equation:
[tex]\[ 8x^3 + 50x^2 - 41x + 7 = 0 \][/tex]
1. Identify the constant term and its factors:
The constant term is [tex]\( 7 \)[/tex]. The factors of [tex]\( 7 \)[/tex] are:
[tex]\[
\pm 1, \pm 7
\][/tex]
2. Identify the leading coefficient and its factors:
The leading coefficient is [tex]\( 8 \)[/tex]. The factors of [tex]\( 8 \)[/tex] are:
[tex]\[
\pm 1, \pm 2, \pm 4, \pm 8
\][/tex]
3. Formulate all possible rational roots:
According to the Rational Zero Theorem, the possible rational roots are given by all possible values of [tex]\( \frac{p}{q} \)[/tex], where [tex]\( p \)[/tex] is a factor of the constant term ([tex]\( \pm 1, \pm 7 \)[/tex]) and [tex]\( q \)[/tex] is a factor of the leading coefficient ([tex]\( \pm 1, \pm 2, \pm 4, \pm 8 \)[/tex]).
Combining these, we get the possible rational roots:
[tex]\[
\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{8}, \pm 7, \pm \frac{7}{2}, \pm \frac{7}{4}, \pm \frac{7}{8}
\][/tex]
Thus, listing all the possible rational roots, we have:
[tex]\[
\{\pm 1, \pm 7, \pm \frac{1}{2}, \pm \frac{7}{2}, \pm \frac{1}{4}, \pm \frac{7}{4}, \pm \frac{1}{8}, \pm \frac{7}{8}\}
\][/tex]
Given the multiple-choice options:
A. [tex]\(\pm 1, \pm 7\)[/tex]
B. [tex]\(\pm 1, \pm 2, \pm 4, \pm 8\)[/tex]
C. [tex]\(\pm 1, \pm 2, \pm 4, \pm 8, \pm \frac{1}{7}, \pm \frac{2}{7}, \pm \frac{4}{7}, \pm \frac{8}{7}\)[/tex]
D. [tex]\(\pm 1, \pm 7, \pm \frac{1}{2}, \pm \frac{7}{2}, \pm \frac{1}{4}, \pm \frac{7}{4}, \pm \frac{1}{8}, \pm \frac{7}{8}\)[/tex]
The correct answer is:
D. [tex]\(\pm 1, \pm 7, \pm \frac{1}{2}, \pm \frac{7}{2}, \pm \frac{1}{4}, \pm \frac{7}{4}, \pm \frac{1}{8}, \pm \frac{7}{8}\)[/tex]
