How do I simplify this rational expression?

Question

(4x+1)/(x—8) — (2)/(x+2) — (8x+16)/(x^2—6x—16) 

Answer 1

Answer: (4x²-x+2)/(x²-6x-16)

 

Explanation:

You want to find a common denominator between all the terms so you can do addition and subtraction of the terms on top to simplify everything. 

 

The main hint at the common denominator what's currently written; the terms (x-8) and (x+2) likely multiply to give you (x²-6x-16), in the third term, which through factorization you see. In other words, decomposing (x²-6x-16) will give you (x-8) and (x+2). 

 

So now that you know that each term should have (x-8)(x+2) in it, you need to multiply the top and bottom of each fraction by the missing term. So for the first, (x+2) is missing and the second (x-8) is missing. The third neither is missing, so it can be left alone. 

 

Now that everything is written with the same denominator (x-8)(x+2), or (x²-6x-16) depending how you've written it, combined all the top terms. I'll write what you have before simplification and after simplification:

 

Before --> (4x+1)(x+2) - 2(x-8) - 8x-16 / (x-8)(x+2)

After --> 4x²-x+2 /  (x²-6x-16) <-- final answer

 

Note: In the final answer, you want to check that the terms cannot be factored and terms canceled on top and bottomfor even further simplification. I tried to simplify further and I couldn't.

 

Hope this helps,

Sean 

Answer 2

(x - 8)(x + 2) = x² - 6x - 16.

 

 

(4x + 1)/(x - 8) - 2/(x + 2) - (8x + 16)/(x² - 6x - 16)

 

= (4x + 1)(x + 2)/(x² - 6x - 16) - 2(x + 8)/(x² - 6x - 16) - (8x + 16)/(x² - 6x - 16)

 

= [(4x + 1)(x + 2) - 2(x + 8) - (8x + 16)]/(x² - 6x - 16)

 

= (4x² + 9x + 2 - 2x - 16 - 8x - 16)/(x² - 6x - 16)

 

= (4x² - x - 30)/(x² - 8x - 16)