To solve the given problem, let's analyze each part step-by-step.
### (a) Listing all possible rational roots according to the Rational Zero Theorem
According to the Rational Zero Theorem, any potential rational root of the polynomial equation [tex]\(8x^3 + 50x^2 - 41x + 7 = 0\)[/tex] is of the form [tex]\(\frac{p}{q}\)[/tex], where:
- [tex]\(p\)[/tex] is a factor of the constant term (7)
- [tex]\(q\)[/tex] is a factor of the leading coefficient (8)
Factors of the constant term 7:
[tex]\[
\pm 1, \pm 7
\][/tex]
Factors of the leading coefficient 8:
[tex]\[
\pm 1, \pm 2, \pm 4, \pm 8
\][/tex]
Possible rational roots [tex]\(\frac{p}{q}\)[/tex]:
[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]
Therefore, 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]
### (b) Identifying an actual root using synthetic division
To find an actual root, we test the possible rational roots listed in part (a) using synthetic division or direct substitution into the polynomial. For simplicity, synthetic division systematically checks whether substituting each root results in zero.
Testing the rational roots:
- Substituting [tex]\(x = -7\)[/tex]:
[tex]\[ 8(-7)^3 + 50(-7)^2 - 41(-7) + 7 = 8(-343) + 50(49) + 287 + 7 \][/tex]
[tex]\[ = -2744 + 2450 + 287 + 7 \][/tex]
[tex]\[ = 0 \][/tex]
Since substituting [tex]\(x = -7\)[/tex] yields zero, it means that [tex]\(x = -7\)[/tex] is a root of the polynomial.
Therefore, one rational root of the given equation is:
[tex]\[
-7
\][/tex]
