Question

How do you simplify `(x^2 + 5x + 4 )/ (x^2 - 16)`?

Answer 1

`(x+1)/(x-4)," ""if "x+4!= 0`

Explanation:

Step 1: Factor both the numerator and denominator.

`(x^2 + 5x + 4)/(x^2 - 16) = [(x+1)(x+4)]/[(x-4)(x+4)]`

Step 2: Cancel common factors.

`[(x+1)cancel((x+4))]/[(x-4)cancel((x+4))]= (x+1)/(x-4)," ""if "x+4!= 0`

Why write this "if `x+4!=0`" part? Because in the original quotient, if `x+4=0` (that is, if `x=–4`), then the denominator would equal zero, and division by zero is undefined.

On its own, `(x+1)/(x-4)` does not give the same problem when `x=–4.` Since we want our simplified expression to reflect the original one in its entirety, we make sure it has all the same restrictions, including `x+4!=0.`