the complete question in the attached figure
we know that
The Rational Root Theorem establishes that if the polynomial
P(x) = a n x ^(n) + a n – 1 x ^(n – 1) + ... + a2 x² + a1 x + a 0
has any rational roots, then they must be of the form
+/-{factors a0/factors an}
then
case A) an=24 a0=-28
The number 24 has factors of +/-{24,12,8,6,4,3,2,1}
The number -28 has factors of +/-{28,14,7,4,2,1}
in this case -7/8 is a potential rational root
case B) an=28 a0=-24
The number 28 has factors of +/-{28,14,7,4,2,1}
The number -24 has factors of +/-{24,12,8,6,4,3,2,1}
in this case -7/8 is not a potential rational root
case C) an=30 a0=-56
The number 30 has factors of +/-{30,15,10,6,5,3,2,1}
The number -56 has factors of +/-{56,28,14,8,7,4,2,1}
in this case -7/8 is not a potential rational root
case D) an=56 a0=-30
The number 56 has factors of +/-{56,28,14,8,7,4,2,1}
The number -30 has factors of +/-{30,15,10,6,5,3,2,1}
in this case -7/8 is not a potential rational root
the answer is
-7/8 is a potential rational root of
f(x) = 24x7 + 3x6 + 4x3 – x – 28
