Question

What is the sum?

StartFraction 3 Over x EndFraction + StartFraction 4 Over x squared EndFraction
StartFraction 3 x + 4 Over x cubed EndFraction
StartFraction 7 Over x EndFraction
StartFraction 3 x + 4 Over x squared EndFraction
StartFraction 7 Over x + x squared EndFraction

Answer 1

Interpreting the terms as
3/x + 4/x^2 + (3x+4)/x^3 + 7/x + (3x+4)/x^2 + 7/(x+x^2),
combine like terms: 3/x + 7/x = 10/x and 4/x^2 + (3x+4)/x^2 = (3x+8)/x^2. So the sum is

10/x + (3x+8)/x^2 + (3x+4)/x^3 + 7/(x(1+x)).

Put over the common denominator x^3(1+x):

Numerator = 10x^2(1+x) + x(1+x)(3x+8) + (3x+4)(1+x) + 7x^2
= 10x^3+10x^2 + 3x^3+11x^2+8x + 3x^2+7x+4 + 7x^2
= 13x^3+31x^2+15x+4.

Thus the sum is
(13x^3 + 31x^2 + 15x + 4) / (x^3(1+x)),

with domain x ≠ 0, x ≠ −1.