Finding the Centroid and Area of Triangle ABC Given Coordinates

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.

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 $f(x)$

Let $f(x) = x^3 + 2x^2 + mx + n$. When $f(x)$ is divided by $(x - 2)$ and $(x - 3)$, the remainders are 33 and 70 respectively. Find the values of $m$ and $n$