Question

What is the difference in simplest form? (n^2+10n+21/n^2+3n-28)-(3n/n-4)

Answer 1

The answer is
`(-2n+3)/(n-4)`

Answer 2

`(3-2n)/(n-4)`

Explanation:

We have been given the expression

`(n^2+10n+21)/(n^2+3n-28)-(3n)/(n-4)`

Let us write the numerator and denominator of first expression in factored form using AC method.

`n^2+10n+21\\=n^2+7n+3n+21\\n(n+7)+3(n+7)\\(n+7)(n+3)`

And the factors of denominator is

`n^2+3n-28\\=n^2+7n-4n-28\\n(n+7)-4(n+7)\\(n+7)(n-4)`

Therefore, the expression becomes

`((n+7)(n+3))/((n+7)(n-4))-(3n)/(n-4)\\\\\text{Cancel the common terms}\\\\(n+3)/(n-4)-(3n)/(n-4)\\\\\text{Denominator of both rational expression is same}\\\text{hence, we can directly add the numerators}\\\\(n+3-3n)/(n-4)\\\\(3-2n)/(n-4)`

Thus, the simplified form is
`(3-2n)/(n-4)`