Question

(x-5) divided by (x^2+3x-40)

Answer 1

`(1)/(x+8)`

Explanation:

Here is your original equation:

`(x-5)/(x^(2) +3x-40)`

What we need to focus on is simplification, specifically simplifying the denominator. First, let's isolate the denominator (we'll get back to
`x-5` later):

`x^(2) +3x-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
`x^(2)` has an exponent of 2. When variables are multiplied, the exponents are added, meaning that
`x*x=x^(2)`. (Or
`x^(1) + x^(1) =x^(2)` , 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-)`

Next, let's look at the factors of 40:

1 and 40 2 and 20 4 and 10 5 and 8

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!

`8-5=3`

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
`3x`. Therefore,
`8(-5)=-40`.

`x^(2) +3x-40 = (x+8)(x-5)`

Now let's go back to our original equation and substitute our old expression with the new one:

`(x-5)/((x+8)(x-5))`

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
`(1)/(1)`.

`(x-5)/((x+8)(x-5))` =
`(1)/(x+8)`

Therefore, your simplified answer is
`(1)/(x+8)` !

Here's the whole process:

`(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)\\\\(x-5)/((x+8)(x-5))=(1)/(x+8) \\\\(1)/(x+8)`

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!

Answer 2

1/ x +8

*The 1 over x + 8