Answer:
![p(x) = {x}^{3} + 4{x}^{2} + 22x- 68](https://tex.z-dn.net/?f=p%28x%29%20%3D%20%7Bx%7D%5E%7B3%7D%20%20%2B%204%7Bx%7D%5E%7B2%7D%20%20%2B%2022x-%2068)
Step-by-step explanation:
We want to write an equation for the polynomial function with roots;
![x = 2](https://tex.z-dn.net/?f=x%20%3D%202)
and
![x = - 3 + 5i](https://tex.z-dn.net/?f=x%20%3D%20%20-%203%20%2B%205i)
Note that complex roots come in pairs and in conjugates.
Therefore
![x = - 3 - 5i](https://tex.z-dn.net/?f=x%20%3D%20%20-%203%20-%205i)
is also a root.
By the factor theorem:
(x-2), (x+3+5i), and (x+3-5i) are factors of the polynomial.
Let p(x) be the polynomial, then in factored form.
![p(x) = (x - 2)(x + 3 + 5i)(x + 3 - 5i)](https://tex.z-dn.net/?f=p%28x%29%20%3D%20%28x%20-%202%29%28x%20%2B%203%20%2B%205i%29%28x%20%2B%203%20-%205i%29)
We expand now to get:
![p(x) = (x - 2)( {x}^{2} + 3x - 5ix + 3x + 9 - 15i + 5ix + 15i + 25)](https://tex.z-dn.net/?f=p%28x%29%20%3D%20%28x%20-%202%29%28%20%7Bx%7D%5E%7B2%7D%20%20%2B%203x%20%20-%20%205ix%20%2B%203x%20%2B%209%20-%2015i%20%2B%205ix%20%2B%2015i%20%2B%2025%29)
Simplify now
![p(x) = (x - 2)( {x}^{2} + 6x +34)](https://tex.z-dn.net/?f=p%28x%29%20%3D%20%28x%20-%202%29%28%20%7Bx%7D%5E%7B2%7D%20%20%2B%206x%20%2B34%29)
We expand further;
![p(x) = x( {x}^{2} + 6x + 34) - 2( {x}^{2} + 6x + 34)](https://tex.z-dn.net/?f=p%28x%29%20%3D%20x%28%20%7Bx%7D%5E%7B2%7D%20%20%2B%206x%20%2B%2034%29%20-%202%28%20%7Bx%7D%5E%7B2%7D%20%20%2B%206x%20%2B%2034%29)
![p(x) = {x}^{3} + 6 {x}^{2} + 34x - 2 {x}^{2} - 12x - 68](https://tex.z-dn.net/?f=p%28x%29%20%3D%20%7Bx%7D%5E%7B3%7D%20%20%2B%206%20%7Bx%7D%5E%7B2%7D%20%20%2B%2034x%20-%202%20%7Bx%7D%5E%7B2%7D%20%20%20-%2012x%20%20-%2068)
![p(x) = {x}^{3} + 4{x}^{2} + 22x- 68](https://tex.z-dn.net/?f=p%28x%29%20%3D%20%7Bx%7D%5E%7B3%7D%20%20%2B%204%7Bx%7D%5E%7B2%7D%20%20%2B%2022x-%2068)
40% of 50 is 20. So, 20/50 students who take the random sample prefer chocolate over vanilla ice cream. In order to find the number of students who prefer chocolate over vanilla ice cream out of 200 students in total, it is best to set up a proportion. Here: 20/50=x/200. Once you solve the proportion you will get x as 80.
Hope the explanation helped!
Answer:
your answer is 8320 i took a quiz with this question and wrote the answer down hope this helps have a great day -cr8855
Step-by-step explanation:
Answer:
1.25 + 5.25
Step-by-step explanation:
combine the similar stuff
(2.75j - 1.5j) +(2.25 + 3)
1.25j + 5.25