1. We can conclude that the result of the division is
![x-13](https://tex.z-dn.net/?f=x-13)
with a remainder of
![4x-9](https://tex.z-dn.net/?f=4x-9)
, or in other notation:
![x-13+ \frac{4x-9}{x^2+3}](https://tex.z-dn.net/?f=x-13%2B%20%5Cfrac%7B4x-9%7D%7Bx%5E2%2B3%7D%20)
. Check the steps in the first image.
3. We can conclude that the result of the division is
![x^{2} +6x+3](https://tex.z-dn.net/?f=x%5E%7B2%7D%20%2B6x%2B3)
with a remainder of 2, or in other notation:
![x^{2}+6x+3+ \frac{2}{x-1}](https://tex.z-dn.net/?f=%20x%5E%7B2%7D%2B6x%2B3%2B%20%5Cfrac%7B2%7D%7Bx-1%7D)
. Check the steps in the firts image.
5. We can conclude that the result of the division is
![x+1](https://tex.z-dn.net/?f=x%2B1)
with a remainder of 5, or in other notation:
![x+1+ \frac{5}{x-4}](https://tex.z-dn.net/?f=x%2B1%2B%20%5Cfrac%7B5%7D%7Bx-4%7D)
. Check the steps in the first image.
7. To check if
![x+4](https://tex.z-dn.net/?f=x%2B4)
is a factor of
![x^{3}+3x^{2}-10x-24](https://tex.z-dn.net/?f=x%5E%7B3%7D%2B3x%5E%7B2%7D-10x-24)
, we are going to perform long division. If the result of the long division has no remainder,
![x+4](https://tex.z-dn.net/?f=x%2B4)
is a factor of
![x^{3}+3x^{2}-10x-24](https://tex.z-dn.net/?f=x%5E%7B3%7D%2B3x%5E%7B2%7D-10x-24)
.
Since the result of the long division has no remainder, we can conclude that
![x+4](https://tex.z-dn.net/?f=x%2B4)
is a factor of
![x^{3}+3x^{2}-10x-24](https://tex.z-dn.net/?f=x%5E%7B3%7D%2B3x%5E%7B2%7D-10x-24)
. Check the steps in the second image.
9. To check if
![x-3](https://tex.z-dn.net/?f=x-3)
is a factor of
![x^{3}+3x^{2}-10x-24](https://tex.z-dn.net/?f=x%5E%7B3%7D%2B3x%5E%7B2%7D-10x-24)
, we are going to perform long division. If the result of the long division has no remainder,
![x-4](https://tex.z-dn.net/?f=x-4)
is a factor of
![x^{3}+3x^{2}-10x-24](https://tex.z-dn.net/?f=x%5E%7B3%7D%2B3x%5E%7B2%7D-10x-24)
.
Since the result of the long division has no remainder, we can conclude that
![x-4](https://tex.z-dn.net/?f=x-4)
is a factor of
![x^{3}+3x^{2}-10x-24](https://tex.z-dn.net/?f=x%5E%7B3%7D%2B3x%5E%7B2%7D-10x-24)
. Check the steps in the third image.
10. We can conclude that the result of the synthetic division is
![-2x^{2}+9x+5](https://tex.z-dn.net/?f=-2x%5E%7B2%7D%2B9x%2B5)
. Check the steps in the fourth image.