According to the Rational Root Theorem, [tex]$-\frac{7}{8}$[/tex] is a potential rational root of which function?

A. [tex]$f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28$[/tex]
B. [tex]$f(x) = 28x^7 + 3x^6 + 4x^3 - x - 24$[/tex]
C. [tex]$f(x) = 30x^7 + 3x^6 + 4x^3 - x - 56$[/tex]
D. [tex]$f(x) = 56x^7 + 3x^6 + 4x^3 - x - 30$[/tex]



Answer :

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].