Perform the following operations and prove closure. Show your work.

Question

x over x+3 + x+2 over x+5

x+4 over x^2+5x+6 • x+3 over x^2−16

2 over x^2−9 − 3x over x^2−5x+6

x+4 over x^2−5x+6 / x^2−16 over x+3

Compare and contrast division of integers to division of rational expressions.

Answer 1

Next time please type your problems in a more normal way

I will do the 3rd:
2/(x^2−9) − 3x/(x^2−5x+6)

step1: try to factor
= 2/((x+3)(x-3)) - 3x/((x-2)(x-3))
so the LCD is (x+3)(x-3)(x-2)
= (2(x-2) - 3x(x+3) )/( (x+3)(x-3)(x-2))
= (2x - 4 - 3x^2 - 3x) / ( (x+3)(x-3)(x-2))
= -(3x^2 + x + 4) / ( (x+3)(x-3)(x-2))

there should have been a restriction stated throughout, namely
x ≠ ± 3, 2