To determine if [tex]\(-\frac{7}{8}\)[/tex] is a potential rational root for any of the given polynomials, we need to substitute [tex]\( x = -\frac{7}{8} \)[/tex] into each polynomial and evaluate the result. If the value of the polynomial at [tex]\( x = -\frac{7}{8} \)[/tex] is zero, then [tex]\(-\frac{7}{8}\)[/tex] is indeed a root of that polynomial.
Consider each polynomial one by one:
1. First polynomial: [tex]\( f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28 \)[/tex]
[tex]\[
f\left(-\frac{7}{8}\right) = 24\left(-\frac{7}{8}\right)^7 + 3\left(-\frac{7}{8}\right)^6 + 4\left(-\frac{7}{8}\right)^3 - \left(-\frac{7}{8}\right) - 28
\][/tex]
The result is approximately [tex]\(-37.883\)[/tex], which is not zero.
2. Second polynomial: [tex]\( f(x) = 28x^7 + 3x^6 + 4x^3 - x - 24 \)[/tex]
[tex]\[
f\left(-\frac{7}{8}\right) = 28\left(-\frac{7}{8}\right)^7 + 3\left(-\frac{7}{8}\right)^6 + 4\left(-\frac{7}{8}\right)^3 - \left(-\frac{7}{8}\right) - 24
\][/tex]
The result is approximately [tex]\(-35.454\)[/tex], which is not zero.
3. Third polynomial: [tex]\( f(x) = 30x^7 + 3x^6 + 4x^3 - x - 56 \)[/tex]
[tex]\[
f\left(-\frac{7}{8}\right) = 30\left(-\frac{7}{8}\right)^7 + 3\left(-\frac{7}{8}\right)^6 + 4\left(-\frac{7}{8}\right)^3 - \left(-\frac{7}{8}\right) - 56
\][/tex]
The result is approximately [tex]\(-68.239\)[/tex], which is not zero.
4. Fourth polynomial: [tex]\( f(x) = 56x^7 + 3x^6 + 4x^3 - x - 30 \)[/tex]
[tex]\[
f\left(-\frac{7}{8}\right) = 56\left(-\frac{7}{8}\right)^7 + 3\left(-\frac{7}{8}\right)^6 + 4\left(-\frac{7}{8}\right)^3 - \left(-\frac{7}{8}\right) - 30
\][/tex]
The result is approximately [tex]\(-52.449\)[/tex], which is not zero.
Since the value of the polynomial at [tex]\( x = -\frac{7}{8} \)[/tex] is not zero for any of the four polynomials, we conclude that [tex]\(-\frac{7}{8}\)[/tex] is not a rational root for any of the given functions.
