Answer:
![\frac{1}{x+8}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bx%2B8%7D)
Step-by-step explanation:
Here is your original equation:
<h2>
![\frac{x-5}{x^{2} +3x-40}](https://tex.z-dn.net/?f=%5Cfrac%7Bx-5%7D%7Bx%5E%7B2%7D%20%2B3x-40%7D)
</h2>
What we need to focus on is simplification, specifically simplifying the denominator. First, let's isolate the denominator (we'll get back to
later):
![x^{2} +3x-40](https://tex.z-dn.net/?f=x%5E%7B2%7D%20%2B3x-40)
Next, we take a look at -40. Our goal is to find what factors of 40 create a sum of 3 so we can separate the equation for polynomial division.
Let's begin building our expressions. We know that
has an exponent of 2. When variables are multiplied, the exponents are added, meaning that
. (Or
, the 1s just don't show)
We can also see that -40 is a negative number. A negative multiplied by a positive makes a negative product, so we can also include one positive and one negative in each expression.
![x^{2} +3x-40 = (x+)(x-)](https://tex.z-dn.net/?f=x%5E%7B2%7D%20%2B3x-40%20%3D%20%28x%2B%29%28x-%29)
Next, let's look at the factors of 40:
<em>1 and 40 2 and 20 4 and 10 5 and 8</em>
Which of these factors can make a sum of 3? Remember that one number has to be positive and the other has to be negative!
<h2>
![8-5=3](https://tex.z-dn.net/?f=8-5%3D3)
</h2>
In this instance, our positive number is 8 and our negative number is 5. They create the sum of 3 that we're splitting in
. Therefore,
.
<h2>
![x^{2} +3x-40 = (x+8)(x-5)](https://tex.z-dn.net/?f=x%5E%7B2%7D%20%2B3x-40%20%3D%20%28x%2B8%29%28x-5%29)
</h2><h2 />
Now let's go back to our original equation and substitute our old expression with the new one:
<h2>
![\frac{x-5}{(x+8)(x-5)}](https://tex.z-dn.net/?f=%5Cfrac%7Bx-5%7D%7B%28x%2B8%29%28x-5%29%7D)
</h2>
Do you notice that the numerator and part of the denominator are equal? This means that they can cancel each other out! Think of it as a
.
<h2>
![\frac{x-5}{(x+8)(x-5)}](https://tex.z-dn.net/?f=%5Cfrac%7Bx-5%7D%7B%28x%2B8%29%28x-5%29%7D)
=
![\frac{1}{x+8}](https://tex.z-dn.net/?f=%5Cfrac%7B1%7D%7Bx%2B8%7D)
</h2>
Therefore, your simplified answer is
!
Here's the whole process:
![\frac{x-5}{x^{2} +3x-40} \\\\(x+?)(x-?)\\\\40:\\1, 40\\2, 20\\4, 10\\5, 8\\\\8-5=3 \\(8x-5x=3x)\\\\(x+8)(x-5)\\\\\frac{x-5}{(x+8)(x-5)}=\frac{1}{x+8} \\\\\frac{1}{x+8}](https://tex.z-dn.net/?f=%5Cfrac%7Bx-5%7D%7Bx%5E%7B2%7D%20%2B3x-40%7D%20%5C%5C%5C%5C%28x%2B%3F%29%28x-%3F%29%5C%5C%5C%5C40%3A%5C%5C1%2C%2040%5C%5C2%2C%2020%5C%5C4%2C%2010%5C%5C5%2C%208%5C%5C%5C%5C8-5%3D3%20%5C%5C%288x-5x%3D3x%29%5C%5C%5C%5C%28x%2B8%29%28x-5%29%5C%5C%5C%5C%5Cfrac%7Bx-5%7D%7B%28x%2B8%29%28x-5%29%7D%3D%5Cfrac%7B1%7D%7Bx%2B8%7D%20%5C%5C%5C%5C%5Cfrac%7B1%7D%7Bx%2B8%7D)
To check your answer, confirm your expressions using the box method attached.
I hope this helps! If you have any more questions or concerns about the answer or the process, let me know!