To determine for which function [tex]\(-\frac{7}{8}\)[/tex] is a potential rational root, we will apply the Rational Root Theorem. This theorem states that any rational root, in its simplest form [tex]\(\frac{p}{q}\)[/tex], must satisfy the condition that [tex]\(p\)[/tex] is a factor of the constant term (the term without [tex]\(x\)[/tex]) and [tex]\(q\)[/tex] is a factor of the leading coefficient (the coefficient of the term with the highest degree).
Given that [tex]\(-\frac{7}{8}\)[/tex] is a potential rational root, we identify [tex]\(p = -7\)[/tex] and [tex]\(q = 8\)[/tex].
We need to check each provided function:
1. For [tex]\(f(x) = 56x^7 + 3x^6 + 4x^3 - x - 30\)[/tex]:
- Constant term: [tex]\(-30\)[/tex]
- Leading coefficient: [tex]\(56\)[/tex]
Checking if [tex]\(7\)[/tex] (the absolute value of [tex]\(p\)[/tex]) is a factor of [tex]\(-30\)[/tex]:
[tex]\[
-30 \div 7 \neq \text{integer} \quad (\text{so } 7 \text{ is not a factor of } -30)
\][/tex]
2. For [tex]\(f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28\)[/tex]:
- Constant term: [tex]\(-28\)[/tex]
- Leading coefficient: [tex]\(24\)[/tex]
Checking if [tex]\(7\)[/tex] is a factor of [tex]\(-28\)[/tex]:
[tex]\[
-28 \div 7 = -4 \quad (\text{7 is a factor of } -28)
\][/tex]
Checking if [tex]\(8\)[/tex] is a factor of [tex]\(24\)[/tex]:
[tex]\[
24 \div 8 = 3 \quad (\text{8 is a factor of 24})
\][/tex]
Since both conditions are met, [tex]\(-\frac{7}{8}\)[/tex] is a potential rational root for this function.
3. For [tex]\(f(x) = 30x^7 + 3x^6 + 4x^3 - x - 56\)[/tex]:
- Constant term: [tex]\(-56\)[/tex]
- Leading coefficient: [tex]\(30\)[/tex]
Checking if [tex]\(7\)[/tex] is a factor of [tex]\(-56\)[/tex]:
[tex]\[
-56 \div 7 = -8 \quad (\text{7 is a factor of } -56)
\][/tex]
Checking if [tex]\(8\)[/tex] is a factor of [tex]\(30\)[/tex]:
[tex]\[
30 \div 8 \neq \text{integer} \quad (\text{so } 8 \text{ is not a factor of } 30)
\][/tex]
4. For [tex]\(f(x) = 28x^7 + 3x^6 + 4x^3 - x - 24\)[/tex]:
- Constant term: [tex]\(-24\)[/tex]
- Leading coefficient: [tex]\(28\)[/tex]
Checking if [tex]\(7\)[/tex] is a factor of [tex]\(-24\)[/tex]:
[tex]\[
-24 \div 7 \neq \text{integer} \quad (\text{7 is not a factor of } -24)
\][/tex]
Based on these evaluations, [tex]\(-\frac{7}{8}\)[/tex] can be a potential rational root only for the function [tex]\(f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28\)[/tex].
Hence, according to the Rational Root Theorem, [tex]\(-\frac{7}{8}\)[/tex] is a potential rational root of the second function:
[tex]\[
f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28
\][/tex]
