First Principles AI

Solution

\frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}

Since degrees are equal, divide first.

\frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}=1+\frac{R(x)}{(x-4)(x-5)(x-6)}

Now let:

\frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}=1+\frac{A}{x-4}+\frac{B}{x-5}+\frac{C}{x-6}

Solving gives $A=3$ , $B=-24$ , $C=30$ .

\boxed{\frac{(x-1)(x-2)(x-3)}{(x-4)(x-5)(x-6)}=1+\frac{3}{x-4}-\frac{24}{x-5}+\frac{30}{x-6}}