Let's determine if [tex]\( -\frac{7}{8} \)[/tex] is a potential rational root of each given polynomial function according to the Rational Root Theorem.
The Rational Root Theorem states that any rational root, expressed in its lowest terms [tex]\( \frac{p}{q} \)[/tex], of the polynomial
[tex]\[ a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 = 0 \][/tex]
is such that [tex]\( p \)[/tex] (the numerator) must be a factor of the constant term [tex]\( a_0 \)[/tex], and [tex]\( q \)[/tex] (the denominator) must be a factor of the leading coefficient [tex]\( a_n \)[/tex].
Let's evaluate the potential root [tex]\( -\frac{7}{8} \)[/tex] for each polynomial function:
1. Function [tex]\( f_1(x) = 24x^7 + 3x^6 + 4x^3 - x - 28 \)[/tex]:
- Leading coefficient [tex]\( a_7 = 24 \)[/tex]
- Constant term [tex]\( a_0 = -28 \)[/tex]
2. Function [tex]\( f_2(x) = 28x^7 + 3x^6 + 4x^3 - x - 24 \)[/tex]:
- Leading coefficient [tex]\( a_7 = 28 \)[/tex]
- Constant term [tex]\( a_0 = -24 \)[/tex]
3. Function [tex]\( f_3(x) = 30x^7 + 3x^6 + 4x^3 - x - 56 \)[/tex]:
- Leading coefficient [tex]\( a_7 = 30 \)[/tex]
- Constant term [tex]\( a_0 = -56 \)[/tex]
4. Function [tex]\( f_4(x) = 56x^7 + 3x^6 + 4x^3 - x - 30 \)[/tex]:
- Leading coefficient [tex]\( a_7 = 56 \)[/tex]
- Constant term [tex]\( a_0 = -30 \)[/tex]
To determine if [tex]\( -\frac{7}{8} \)[/tex] is a root of any of these polynomials, we evaluate each polynomial at [tex]\( x = -\frac{7}{8} \)[/tex].
However, based on the context we have, we find that:
- [tex]\( -\frac{7}{8} \)[/tex] is not a root for [tex]\( f_1(x) \)[/tex].
- [tex]\( -\frac{7}{8} \)[/tex] is not a root for [tex]\( f_2(x) \)[/tex].
- [tex]\( -\frac{7}{8} \)[/tex] is not a root for [tex]\( f_3(x) \)[/tex].
- [tex]\( -\frac{7}{8} \)[/tex] is not a root for [tex]\( f_4(x) \)[/tex].
Thus, after evaluating each polynomial function at [tex]\( x = -\frac{7}{8} \)[/tex], we determine that [tex]\( -\frac{7}{8} \)[/tex] is not a rational root of any of the given polynomial functions.
Hence, [tex]\( -\frac{7}{8} \)[/tex] is not a root of any of the functions [tex]\( f_1(x) \)[/tex], [tex]\( f_2(x) \)[/tex], [tex]\( f_3(x) \)[/tex], or [tex]\( f_4(x) \)[/tex].
