Finding Unknown Coefficients Using Polynomial Division and Remainder

Convert the 16-bit binary number:

into its decimal (base-10) equivalent.

It is given that there exist constants $A$ and $B$ such that

for all $x \in \mathbb{R}$. Find the values of $A$ and $B$. Hence, write down

in partial fractions.

If $A(6, 7)$, $B(12, 9)$, and $C(15, 8)$, find the coordinates of the centroid and the area of triangle ABC

A quadratic polynomial $G(x)$ has remainders $\frac{1}{3}$, $\frac{1}{5}$, and $\frac{1}{8}$ when divided by $(x - 1)$, $(x - 3)$, and $(x - 6)$ respectively. Show that $(x - 1)$, $(x - 3)$, and $(x - 6)$ are factors of the polynomial

Hence, find $f(x)$