Question

According to the Rational Root Theorem, -7/8 is a potential rational root of which function?
A) f(x) = 24x7 + 3x6 + 4x3 – x – 28
B) f(x) = 28x7 + 3x6 + 4x3 – x – 24
C) f(x) = 30x7 + 3x6 + 4x3 – x – 56
D) f(x) = 56x7 + 3x6 + 4x3 – x – 30

Answer 1

Rational Root Theorem:
A.
Factors of P (Constant): 28 : 1, 2, 4, 7, 14, 28
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Factors of Q (Leading Coefficient): 24 : 1, 2, 3, 4, 6, 8, 12, 24
+ - 1, 1/2, 1/3, 1/4, 1/6, 1/8, 1/12, 1/24, 2, 2/3, 4, 4/3, 7, 7/2, 7/3, 7/4, 7/6, 7/8, 7/12, 7/24, 14, 14/3, 28, 28/3,

Answer 2

Rational root theorem is used to determine the possible root of a function.

-7/8 is a potential rational root of $\mathbf{f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28}$

We have:

$\mathbf{f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28}$

The constant term is:

$\mathbf{p=28}$

The factors are:

$\mathbf{p=\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28}$

The leading coefficient is:

$\mathbf{q = 24}$

The factors are:

$\mathbf{q=\pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24}$

So, the possible roots are:

$\mathbf{Roots = \pm \frac{p}{q}}$

One of the factors of p is 7; while one of the factors of q is 8.

So, we have:

$\mathbf{Roots = \pm \frac{7}{8}}$

Hence, -7/8 is a potential rational root of $\mathbf{f(x) = 24x^7 + 3x^6 + 4x^3 - x - 28}$

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