Question

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

Answer 1

Answer:

1/ x +8

*The 1 over x + 8

Answer 2

Answer:

$\frac{1}{x+8}$

Step-by-step explanation:

Here is your original equation:

$\frac{x-5}{x^{2} +3x-40}$

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:

$\frac{x-5}{(x+8)(x-5)}$

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 $\frac{1}{1}$.

$\frac{x-5}{(x+8)(x-5)}$ = $\frac{1}{x+8}$

Therefore, your simplified answer is  $\frac{1}{x+8}$ !

Here's the whole process:

$\frac{x-5}{x^{2} +3x-40} \\\\(x+?)(x-?)\\\\40:\\1, 40\\2, 20\\4, 10\\5, 8\\\\8-5=3 \\(8x-5x=3x)\\\\(x+8)(x-5)\\\\\frac{x-5}{(x+8)(x-5)}=\frac{1}{x+8} \\\\\frac{1}{x+8}$

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!