Answer: The correct option is (B) -17.5.
Step-by-step explanation: We are given to use the Remainder Theorem to find the remainder for the following division :
[tex](-6x^3+3x^2-4)\div(2x-3)[/tex]
Remainder Theorem : When a polynomial p(x) is divided by the linear factor (x - a), then the remainder is equal to p(a).
We have
[tex]2x-3=0\\\\\Rightarrow 2x=3\\\\\Rightarrow x=\dfrac{3}{2}[/tex]
So, the remainder for the given division is given by
[tex]R\\\\=-6\times\left(\dfrac{3}{2}\right)^3+3\times\left(\dfrac{3}{2}\right)^2-4\\\\\\=-6\times\dfrac{27}{8}+3\times\dfrac{9}{4}-4\\\\\\=-\dfrac{81}{4}+\dfrac{27}{4}-4\\\\\\=\dfrac{-81+27-16}{4}\\\\\\=-\dfrac{70}{4}\\\\=-17.5[/tex]
Thus, the required remainder is -17.5.
Option (B) is CORRECT.
