Question

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

(x/x+3) + (x+2/x+5)

Answer 1

`(2(x^(2)+5x+3))/((x+3)(x+5))`

Explanation:

The given expression is
`(x)/(x+3)+(x+2)/(x+5)`

We have to simplify the given expression

`(x)/(x+3)+(x+2)/(x+5)`

=
`(x(x+5)+(x+2)(x+3))/((x+3)(x+5))` [Distributive law]

=
`(x^(2)+5x+x^(2)+3x+2x+6)/((x+3)(x+5))`

=
`(2x^(2)+10x+6)/((x+3)(x+5))`

=
`(2(x^(2)+5x+3))/((x+3)(x+5))`

Finally the simplified form of the given expression is
`(2(x^(2)+5x+3))/((x+3)(x+5))`

Answer 2

`(2(x^(2) + 5x + 3))/((x+3)(x+5))`

Explanation:

We need to sum the following two expressions:

`(x)/(x+3) + (x+2)/(x+5)`

`(x(x+5) + (x+2)(x+3))/((x+3)(x+5))`

expanding the polynomial in the numerator:

`(2x^(2) + 10x + 6)/((x+3)(x+5))`

`(2(x^(2) + 5x + 3))/((x+3)(x+5))`

This is the most simplified form we can get:

`(2(x^(2) + 5x + 3))/((x+3)(x+5))`