when f is the function defied by f(x)= (1-3x)^m / (3-4x)^n,?

Question

find Ax+B so that f(x) = ((1-3x)^m-1) / ((3-4x)^n+1) * ( Ax+B)
holds for all integers m, n >1

A) Ax+B= 4(m+n) x-12n+3m
B) Ax+B= 12(m+n) x+4n +9m
C) Ax+B= 4(m-n) x+12n- 3m
D) Ax+B= 12(m-n) x +4n-9m

Answer 1

It's unclear if the Ax+B is multiplying the numerator or the denominator of the fraction.

We can get f(x) = ((1-3x)^m-1) / ((3-4x)^n+1) by dividing the numerator of the original f(x) by 1-3x and multiplying the denominator of the original f(x) by 3-4x.

This is original f(x) *1/[1-3x)(3-4x)].

Now we are to multiply this by Ax+B and determine A and B such that the new f(x) holds for all m,n >1.

I really don't see how to proceed from here. Nor do I see how trying each of your answers provides the desired result.