By the binomial theorem,
![\displaystyle \left(x^2 + \frac px\right)^6 = \sum_{n=0}^6 \binom 6n \left(x^2\right)^n \left(\frac px\right)^{6-n} = \sum_{n=0}^6 p^{6-n} \binom 6n x^{3n-6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cleft%28x%5E2%20%2B%20%5Cfrac%20px%5Cright%29%5E6%20%3D%20%5Csum_%7Bn%3D0%7D%5E6%20%5Cbinom%206n%20%5Cleft%28x%5E2%5Cright%29%5En%20%5Cleft%28%5Cfrac%20px%5Cright%29%5E%7B6-n%7D%20%3D%20%5Csum_%7Bn%3D0%7D%5E6%20p%5E%7B6-n%7D%20%5Cbinom%206n%20x%5E%7B3n-6%7D)
where
![\dbinom kn = \dfrac{k!}{n!(k-n)!}](https://tex.z-dn.net/?f=%5Cdbinom%20kn%20%3D%20%5Cdfrac%7Bk%21%7D%7Bn%21%28k-n%29%21%7D)
is the so-called binomial coefficient.
The term that's independent of x is the constant term, which occurs when
, or
. Given that this constant 240, we have
![p^{6-2} \dbinom62 = 240 \implies p^4 = 16 \implies p = \pm2](https://tex.z-dn.net/?f=p%5E%7B6-2%7D%20%5Cdbinom62%20%3D%20240%20%5Cimplies%20p%5E4%20%3D%2016%20%5Cimplies%20p%20%3D%20%5Cpm2)
but
must be positive, so ![\boxed{p=2}](https://tex.z-dn.net/?f=%5Cboxed%7Bp%3D2%7D)
A linear regression is a regression that can be represented by a linear line of best fit.
The equation of the line of best fit is ![y = 2.4x +33.2](https://tex.z-dn.net/?f=y%20%3D%202.4x%20%2B33.2)
<h3>How to determine the equation</h3>
From the graph, we have the following ordered pairs
(x,y) = (50,150) and (100,270)
The slope (m) is then calculated as:
![m = \frac{270 - 150}{100 - 50}](https://tex.z-dn.net/?f=m%20%3D%20%5Cfrac%7B270%20-%20150%7D%7B100%20-%2050%7D)
This gives
![m = 2.4](https://tex.z-dn.net/?f=m%20%3D%202.4)
The regression line crosses the y-axis at b = 33.2
A linear regression is represented as:
y = mx + b
So, we have:
![y = 2.4x +33.2](https://tex.z-dn.net/?f=y%20%3D%202.4x%20%2B33.2)
Hence, the equation of the line of best fit is ![y = 2.4x +33.2](https://tex.z-dn.net/?f=y%20%3D%202.4x%20%2B33.2)
Read more about line of best fit at:
brainly.com/question/15827538
To find the answer, just divide 48 by 8, which is 6, so the temperature fell by 6°F each minute.
The solution to the inequality expression is x ≥ 30
<h3>How to solve the
inequality expression?</h3>
The inequality expression is given as:
8x - 3(2x - 4) ≤ 3(x - 6)
Open the brackets in the above inequality expression
8x - 6x + 12 ≤ 3x - 18
Collect the like terms in the above inequality expression
8x - 6x - 3x ≤ -12 - 18
Evaluate the like terms in the above inequality expression
-x ≤ -30
Divide both sides of the above inequality expression by -1
x ≥ 30
Hence, the solution to the inequality expression is x ≥ 30
Read more about inequality expression at
brainly.com/question/24372553
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Answer: A (5, -5.5)
Step-by-step explanation: To find the midpoint of a line segment, you must know the formations of points. Two coordinates are made up of (x1, y1)(x2, y2). Now, to find the answer, you have to add the two x's (which is 4 and 6 = 10) and divide by 2 which equals 5. Now to find the y point, add both y's together. -5 + -6 = -11, and dividing by 2 gives you -5.5
Thus, the midpoint is (5, -5.5)