Question

For the rational expression 2x^2-7x-15/x^2-25, do both of the following,

a. write the rational expression in factored form
b. state any restrictions on the variable

Answer 1

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

b. x ≠ -5 (Vertical asymptote) and x ≠ 5 (Hole)

Explanation:

Factor the numerator (Grouping):

`2x^2 - 7x - 15`

Two numbers that multiply to -30 and add to -7 = -3 and 10

`[2x^2 - 10x] + [3x - 15]`

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

Factor the denominator (Difference of Two Squares):

`x^2 - 25` =
`(x +5)(x-5)`

Factored Expression:

(x - 5) can be factored out of top and bottom as a hole-

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

Variable Restrictions:

Denominator ≠ 0

`x + 5 = 0\\x = -5`

Vertical asymptote at x = -5 ⇒ x ≠ -5