Answer:
x³ - 5x² + 81x - 405 = 0
Step-by-step explanation:
Complex roots occur in conjugate pairs.
Thus given x = - 9i is a root then x = 9i is also a root
The factors are then (x - 5), (x - 9i) and (x + 9i)
The polynomial is the the product of the roots, that is
f(x) = (x - 5)(x - 9i)(x + 9i) ← expand the complex factors
= (x - 5)(x² - 81i²) → note i² = - 1
= (x - 5)(x² + 81) ← distribute
= x³ + 81x - 5x² - 405, thus
x³ - 5x² + 81x - 405 = 0 ← is the polynomial equation
A
(a) You're looking for
![P(X\le 8) = \displaystyle \sum_{x=0}^8 P(X=x)](https://tex.z-dn.net/?f=P%28X%5Cle%208%29%20%3D%20%5Cdisplaystyle%20%5Csum_%7Bx%3D0%7D%5E8%20P%28X%3Dx%29)
where
![P(X=x) = \begin{cases}\dfrac{\lambda^x e^{-\lambda}}{x!}&\text{if }x\in\{0,1,2,\ldots\}\\0&\text{otherwise}\end{cases}](https://tex.z-dn.net/?f=P%28X%3Dx%29%20%3D%20%5Cbegin%7Bcases%7D%5Cdfrac%7B%5Clambda%5Ex%20e%5E%7B-%5Clambda%7D%7D%7Bx%21%7D%26%5Ctext%7Bif%20%7Dx%5Cin%5C%7B0%2C1%2C2%2C%5Cldots%5C%7D%5C%5C0%26%5Ctext%7Botherwise%7D%5Cend%7Bcases%7D)
Customers arrive at a mean rate of 6 customers per 10 minutes, or equivalently 12 customers per 20 minutes, so
![\lambda = \dfrac{12\,\rm customers}{20\,\rm min}\times(20\,\mathrm{min}) = 12\,\mathrm{customers}](https://tex.z-dn.net/?f=%5Clambda%20%3D%20%5Cdfrac%7B12%5C%2C%5Crm%20customers%7D%7B20%5C%2C%5Crm%20min%7D%5Ctimes%2820%5C%2C%5Cmathrm%7Bmin%7D%29%20%3D%2012%5C%2C%5Cmathrm%7Bcustomers%7D)
Then
![\displaystyle P(X\le 8) = \sum_{x=0}^8 \frac{12^x e^{-12}}{x!} \approx \boxed{0.155}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20P%28X%5Cle%208%29%20%3D%20%5Csum_%7Bx%3D0%7D%5E8%20%5Cfrac%7B12%5Ex%20e%5E%7B-12%7D%7D%7Bx%21%7D%20%5Capprox%20%5Cboxed%7B0.155%7D)
(b) Now you want
![P(X\ge4) = 1 - P(X](https://tex.z-dn.net/?f=P%28X%5Cge4%29%20%3D%201%20-%20P%28X%3C4%29%20%3D%201%20-%20%5Cdisplaystyle%5Csum_%7Bx%3D0%7D%5E3%20P%28X%3Dx%29)
This time, we have
![\lambda = \dfrac{6\,\rm customers}{10\,\rm min}\times(10\,\mathrm{min}) = 6\,\mathrm{customers}](https://tex.z-dn.net/?f=%5Clambda%20%3D%20%5Cdfrac%7B6%5C%2C%5Crm%20customers%7D%7B10%5C%2C%5Crm%20min%7D%5Ctimes%2810%5C%2C%5Cmathrm%7Bmin%7D%29%20%3D%206%5C%2C%5Cmathrm%7Bcustomers%7D)
so that
![P(X\ge4) = 1 - \displaystyle \sum_{x=0}^3 \frac{6^x e^{-6}}{x!} \approx \boxed{0.849}](https://tex.z-dn.net/?f=P%28X%5Cge4%29%20%3D%201%20-%20%5Cdisplaystyle%20%5Csum_%7Bx%3D0%7D%5E3%20%5Cfrac%7B6%5Ex%20e%5E%7B-6%7D%7D%7Bx%21%7D%20%5Capprox%20%5Cboxed%7B0.849%7D)
B
(a) In other words, you're asked to find the probability that more than 1 customer shows up in the same minute, or
![P(X > 1) = 1 - P(X \le 1) = 1 - P(X=0) - P(X=1)](https://tex.z-dn.net/?f=P%28X%20%3E%201%29%20%3D%201%20-%20P%28X%20%5Cle%201%29%20%3D%201%20-%20P%28X%3D0%29%20-%20P%28X%3D1%29)
with
![\lambda = \dfrac{6\,\rm customers}{6\,\rm min}\times(1\,\mathrm{min}) = 1\,\mathrm{customer}](https://tex.z-dn.net/?f=%5Clambda%20%3D%20%5Cdfrac%7B6%5C%2C%5Crm%20customers%7D%7B6%5C%2C%5Crm%20min%7D%5Ctimes%281%5C%2C%5Cmathrm%7Bmin%7D%29%20%3D%201%5C%2C%5Cmathrm%7Bcustomer%7D)
So we have
![P(X > 1) = 1 - \dfrac{1^0 e^{-1}}{0!} - \dfrac{1^1 e^{-1}}{1!} \approx \boxed{0.264}](https://tex.z-dn.net/?f=P%28X%20%3E%201%29%20%3D%201%20-%20%5Cdfrac%7B1%5E0%20e%5E%7B-1%7D%7D%7B0%21%7D%20-%20%5Cdfrac%7B1%5E1%20e%5E%7B-1%7D%7D%7B1%21%7D%20%5Capprox%20%5Cboxed%7B0.264%7D)
C
(a) Similar to B, you're looking for
![P(X \le 1) = P(X=0) + P(X=1)](https://tex.z-dn.net/?f=P%28X%20%5Cle%201%29%20%3D%20P%28X%3D0%29%20%2B%20P%28X%3D1%29)
with
![\lambda = \dfrac{12\,\rm customers}{6\,\rm min}\times(1\,\mathrm{min}) = 2\,\mathrm{customers}](https://tex.z-dn.net/?f=%5Clambda%20%3D%20%5Cdfrac%7B12%5C%2C%5Crm%20customers%7D%7B6%5C%2C%5Crm%20min%7D%5Ctimes%281%5C%2C%5Cmathrm%7Bmin%7D%29%20%3D%202%5C%2C%5Cmathrm%7Bcustomers%7D)
so that
![P(X\le1) = \dfrac{2^0e^{-2}}{0!} + \dfrac{2^1e^{-2}}{1!} \approx \boxed{0.406}](https://tex.z-dn.net/?f=P%28X%5Cle1%29%20%3D%20%5Cdfrac%7B2%5E0e%5E%7B-2%7D%7D%7B0%21%7D%20%2B%20%5Cdfrac%7B2%5E1e%5E%7B-2%7D%7D%7B1%21%7D%20%5Capprox%20%5Cboxed%7B0.406%7D)
Answer:
2b+9
Step-by-step explanation:
(7-b)+(3b+2) ---> remove parenthesis
7-b+3b+2 ---> combine like terms
2b+9
Answer:
32'cube
Step-by-step explanation:
Just i have no paper and pen so i just calculated it in my mind so its not practically an accurate answer