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f(x) =x^3 -9x g(x)=x^2 -2x -3 Which...

`f(x) =x^3 -9x`
`g(x)=x^2 -2x -3`
Which of the following expressions is equivalent to `(f(x))/(g(x))`, for x gt 3 ?

A

`1/(x+1)`

B

`(x+3)/(x+1)`

C

`(x^2(x+3))/(x+1)`

D

`(x(x+3))/(x+1)`

Text Solution

Verified by Experts

The correct Answer is:
D
Since `x^3 -9x =x(x+3)(x-3)` and `x^2-2x-3=(x-1)(x-3)` , the fraction `(f(x))/(g(x))` can be written as `(x(+3)(x-3))/((x+1)(x-3))` . It is given that x gt 3, so the common factor x – 3 is not equal to 0. Therefore, the fraction can be further simplified to `(x(x+3))/(x+1)` . Choice A is incorrect. The expression `1/(x+1)` is not equivalent to `(f(x))/(g(x))` because at x=0, `1/(x+1)` as a value of 1 and `(f(x))/(g(x))` has a value 0.
Choice B is incorrect and results from omitting the factor x in the factorization of f (x). Choice C is incorrect and may result from incorrectly factoring g (x) as (x + 1)(x + 3) instead of (x +1)(x – 3).
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