Answer :
To determine whether [tex]\(-\frac{7}{8}\)[/tex] is a potential rational root of any of the given polynomial functions, we can use the Rational Root Theorem. The Rational Root Theorem states that if [tex]\(\frac{p}{q}\)[/tex] is a root of a polynomial, where [tex]\(p\)[/tex] is a factor of the constant term and [tex]\(q\)[/tex] is a factor of the leading coefficient, then [tex]\(\frac{p}{q}\)[/tex] may be a potential rational root of the polynomial.
Let's check each polynomial one by one:
1. [tex]\(f_1(x) = 24x^7 + 3x^6 + 4x^3 - x - 28\)[/tex]
- Leading coefficient: [tex]\(24\)[/tex]
- Constant term: [tex]\(-28\)[/tex]
- The possible values for [tex]\(p\)[/tex] (factors of [tex]\(-28\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28\)[/tex].
- The possible values for [tex]\(q\)[/tex] (factors of [tex]\(24\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\)[/tex].
- Therefore, the possible rational roots [tex]\(\frac{p}{q}\)[/tex] include values like [tex]\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{8}, \pm \frac{7}{8}\)[/tex], etc.
Given that [tex]\(-\frac{7}{8}\)[/tex] is one of the potential rational roots evaluated, we can conclude that [tex]\(-\frac{7}{8}\)[/tex] is a potential root of the polynomial [tex]\(f_1(x)\)[/tex].
2. [tex]\(f_2(x) = 28x^7 + 3x^6 + 4x^3 - x - 24\)[/tex]
- Leading coefficient: [tex]\(28\)[/tex]
- Constant term: [tex]\(-24\)[/tex]
- The possible values for [tex]\(p\)[/tex] (factors of [tex]\(-24\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\)[/tex].
- The possible values for [tex]\(q\)[/tex] (factors of [tex]\(28\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28\)[/tex].
- Therefore, the possible rational roots [tex]\(\frac{p}{q}\)[/tex] include values like [tex]\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{7},\)[/tex], etc.
Upon checking, [tex]\(-\frac{7}{8}\)[/tex] is not a possible rational root based on the factors of the constant term and leading coefficient.
3. [tex]\(f_3(x) = 30x^7 + 3x^6 + 4x^3 - x - 56\)[/tex]
- Leading coefficient: [tex]\(30\)[/tex]
- Constant term: [tex]\(-56\)[/tex]
- The possible values for [tex]\(p\)[/tex] (factors of [tex]\(-56\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56\)[/tex].
- The possible values for [tex]\(q\)[/tex] (factors of [tex]\(30\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30\)[/tex].
- Therefore, the possible rational roots [tex]\(\frac{p}{q}\)[/tex] include values like [tex]\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{5},\)[/tex], etc.
Checking, [tex]\(-\frac{7}{8}\)[/tex] is not a potential rational root based on the factors.
4. [tex]\(f_4(x) = 56x^7 + 3x^6 + 4x^3 - x - 30\)[/tex]
- Leading coefficient: [tex]\(56\)[/tex]
- Constant term: [tex]\(-30\)[/tex]
- The possible values for [tex]\(p\)[/tex] (factors of [tex]\(-30\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30\)[/tex].
- The possible values for [tex]\(q\)[/tex] (factors of [tex]\(56\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56\)[/tex].
- Therefore, the possible rational roots [tex]\(\frac{p}{q}\)[/tex] include values like [tex]\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{7},\)[/tex], etc.
Upon checking again, [tex]\(-\frac{7}{8}\)[/tex] does not appear as a potential rational root based on the factors of the constant term and leading coefficient.
Thus, [tex]\(-\frac{7}{8}\)[/tex] is a potential rational root of the polynomial [tex]\(f_1(x) = 24x^7 + 3x^6 + 4x^3 - x - 28\)[/tex].
Let's check each polynomial one by one:
1. [tex]\(f_1(x) = 24x^7 + 3x^6 + 4x^3 - x - 28\)[/tex]
- Leading coefficient: [tex]\(24\)[/tex]
- Constant term: [tex]\(-28\)[/tex]
- The possible values for [tex]\(p\)[/tex] (factors of [tex]\(-28\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28\)[/tex].
- The possible values for [tex]\(q\)[/tex] (factors of [tex]\(24\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\)[/tex].
- Therefore, the possible rational roots [tex]\(\frac{p}{q}\)[/tex] include values like [tex]\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{4}, \pm \frac{1}{6}, \pm \frac{1}{8}, \pm \frac{7}{8}\)[/tex], etc.
Given that [tex]\(-\frac{7}{8}\)[/tex] is one of the potential rational roots evaluated, we can conclude that [tex]\(-\frac{7}{8}\)[/tex] is a potential root of the polynomial [tex]\(f_1(x)\)[/tex].
2. [tex]\(f_2(x) = 28x^7 + 3x^6 + 4x^3 - x - 24\)[/tex]
- Leading coefficient: [tex]\(28\)[/tex]
- Constant term: [tex]\(-24\)[/tex]
- The possible values for [tex]\(p\)[/tex] (factors of [tex]\(-24\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24\)[/tex].
- The possible values for [tex]\(q\)[/tex] (factors of [tex]\(28\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28\)[/tex].
- Therefore, the possible rational roots [tex]\(\frac{p}{q}\)[/tex] include values like [tex]\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{7},\)[/tex], etc.
Upon checking, [tex]\(-\frac{7}{8}\)[/tex] is not a possible rational root based on the factors of the constant term and leading coefficient.
3. [tex]\(f_3(x) = 30x^7 + 3x^6 + 4x^3 - x - 56\)[/tex]
- Leading coefficient: [tex]\(30\)[/tex]
- Constant term: [tex]\(-56\)[/tex]
- The possible values for [tex]\(p\)[/tex] (factors of [tex]\(-56\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56\)[/tex].
- The possible values for [tex]\(q\)[/tex] (factors of [tex]\(30\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30\)[/tex].
- Therefore, the possible rational roots [tex]\(\frac{p}{q}\)[/tex] include values like [tex]\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{3}, \pm \frac{1}{5},\)[/tex], etc.
Checking, [tex]\(-\frac{7}{8}\)[/tex] is not a potential rational root based on the factors.
4. [tex]\(f_4(x) = 56x^7 + 3x^6 + 4x^3 - x - 30\)[/tex]
- Leading coefficient: [tex]\(56\)[/tex]
- Constant term: [tex]\(-30\)[/tex]
- The possible values for [tex]\(p\)[/tex] (factors of [tex]\(-30\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 3, \pm 5, \pm 6, \pm 10, \pm 15, \pm 30\)[/tex].
- The possible values for [tex]\(q\)[/tex] (factors of [tex]\(56\)[/tex]) are [tex]\(\pm 1, \pm 2, \pm 4, \pm 7, \pm 8, \pm 14, \pm 28, \pm 56\)[/tex].
- Therefore, the possible rational roots [tex]\(\frac{p}{q}\)[/tex] include values like [tex]\(\pm 1, \pm \frac{1}{2}, \pm \frac{1}{4}, \pm \frac{1}{7},\)[/tex], etc.
Upon checking again, [tex]\(-\frac{7}{8}\)[/tex] does not appear as a potential rational root based on the factors of the constant term and leading coefficient.
Thus, [tex]\(-\frac{7}{8}\)[/tex] is a potential rational root of the polynomial [tex]\(f_1(x) = 24x^7 + 3x^6 + 4x^3 - x - 28\)[/tex].
