Answer:
The partial fraction decomposition is
.
Step-by-step explanation:
Partial-fraction decomposition is the process of starting with the simplified answer and taking it back apart, of "decomposing" the final expression into its initial polynomial fractions.
To find the partial fraction decomposition of
:
First, the form of the partial fraction decomposition is
![\frac{- 4 x^{2} + 13 x - 12}{\left(x + 1\right)^{2} \left(x + 2\right)}=\frac{A}{x + 1}+\frac{B}{\left(x + 1\right)^{2}}+\frac{C}{x + 2}](https://tex.z-dn.net/?f=%5Cfrac%7B-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%7D%3D%5Cfrac%7BA%7D%7Bx%20%2B%201%7D%2B%5Cfrac%7BB%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%7D%2B%5Cfrac%7BC%7D%7Bx%20%2B%202%7D)
Write the right-hand side as a single fraction:
![\frac{- 4 x^{2} + 13 x - 12}{\left(x + 1\right)^{2} \left(x + 2\right)}=\frac{\left(x + 1\right)^{2} C + \left(x + 1\right) \left(x + 2\right) A + \left(x + 2\right) B}{\left(x + 1\right)^{2} \left(x + 2\right)}](https://tex.z-dn.net/?f=%5Cfrac%7B-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%7D%3D%5Cfrac%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20C%20%2B%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%28x%20%2B%202%5Cright%29%20A%20%2B%20%5Cleft%28x%20%2B%202%5Cright%29%20B%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%7D)
The denominators are equal, so we require the equality of the numerators:
![- 4 x^{2} + 13 x - 12=\left(x + 1\right)^{2} C + \left(x + 1\right) \left(x + 2\right) A + \left(x + 2\right) B](https://tex.z-dn.net/?f=-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%3D%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20C%20%2B%20%5Cleft%28x%20%2B%201%5Cright%29%20%5Cleft%28x%20%2B%202%5Cright%29%20A%20%2B%20%5Cleft%28x%20%2B%202%5Cright%29%20B)
Expand the right-hand side:
![- 4 x^{2} + 13 x - 12=x^{2} A + x^{2} C + 3 x A + x B + 2 x C + 2 A + 2 B + C](https://tex.z-dn.net/?f=-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%3Dx%5E%7B2%7D%20A%20%2B%20x%5E%7B2%7D%20C%20%2B%203%20x%20A%20%2B%20x%20B%20%2B%202%20x%20C%20%2B%202%20A%20%2B%202%20B%20%2B%20C)
The coefficients near the like terms should be equal, so the following system is obtained:
![\begin{cases} A + C = -4\\3 A + B + 2 C = 13\\2 A + 2 B + C = -12 \end{cases}](https://tex.z-dn.net/?f=%5Cbegin%7Bcases%7D%20A%20%2B%20C%20%3D%20-4%5C%5C3%20A%20%2B%20B%20%2B%202%20C%20%3D%2013%5C%5C2%20A%20%2B%202%20B%20%2B%20C%20%3D%20-12%20%5Cend%7Bcases%7D)
Solving this system, we get that
.
Therefore,
![\frac{- 4 x^{2} + 13 x - 12}{\left(x + 1\right)^{2} \left(x + 2\right)}=\frac{50}{x + 1}+\frac{-29}{\left(x + 1\right)^{2}}+\frac{-54}{x + 2}](https://tex.z-dn.net/?f=%5Cfrac%7B-%204%20x%5E%7B2%7D%20%2B%2013%20x%20-%2012%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%20%5Cleft%28x%20%2B%202%5Cright%29%7D%3D%5Cfrac%7B50%7D%7Bx%20%2B%201%7D%2B%5Cfrac%7B-29%7D%7B%5Cleft%28x%20%2B%201%5Cright%29%5E%7B2%7D%7D%2B%5Cfrac%7B-54%7D%7Bx%20%2B%202%7D)
Answer = 4cm^2
8/4=2 2X2=4cm^2
Answer:
yes it does not curve at any point
Answer:
x = 3, y = 4 and z = -5.
Step-by-step explanation:
Use the elimination method:
-3x + 2y + z = -6...........(1)
x + 3y + 2z = 5..............(2)
4x + 4y + 3z = 13..........(3)
Multiply the first equation by -2:
6x - 4y - 2z = 12 Add this to equation (2):
7x - y = 17...........(4)
Now Multiply equation (2) by 3 and equation (3) by -2
3x + 9y + 6z = 15 ............(5)
-8x - 8y - 6z = -26............(6) Adding (5) + (6):
-5x + y = -11..........(7)
Now Add (7) and (4):
2x = 6
x = 3.
Now substitute for x in (4):
7(3) - y = 17
-y = 17 - 21 = -4
y = 4.
Finally substitute for x and y in equation (3) to find z:
4(3) + 4(4) +3z = 13
3z = 13 - 12 - 16 = -15
z = -5.
Answer:
thx for the points
Step-by-step explanation: