Answer:
<h2>see below</h2>
Step-by-step explanation:
<h3>to understand this</h3><h3>you need to know about:</h3>
<h3>given:</h3>
- f(x)=3x³-2x²-12x+8
- -a×-b=ab
- -a×b=-ab
- a²-b²=(a+b)(a-b)
<h3>to do:</h3>
- use factor theorem to show that (x+2) is a factor of f(x)
<h3>tips and formulas:</h3>
- When f(c)=0 then x−c is a factor of f(x)
<h3>let's solve:</h3>
according to the question
x+2=0
x=-2
![\sf substitute \: the \: value \: of \: x \: : \\ \sf 3 {( - 2)}^{3} - 2( { - 2)}^{2} - 12( - 2) + 8](https://tex.z-dn.net/?f=%20%5Csf%20substitute%20%5C%3A%20the%20%5C%3A%20value%20%5C%3A%20of%20%5C%3A%20x%20%5C%3A%20%20%3A%20%20%5C%5C%20%5Csf%203%20%7B%28%20-%202%29%7D%5E%7B3%7D%20-%202%28%20%7B%20-%202%29%7D%5E%7B2%7D%20%20-%2012%28%20-%202%29%20%2B%208)
![\sf simplify \: exponets : \\ 3( - 8) - 2(4) - 12( - 2) + 8](https://tex.z-dn.net/?f=%20%5Csf%20simplify%20%5C%3A%20exponets%20%3A%20%20%5C%5C%203%28%20-%208%29%20-%202%284%29%20-%2012%28%20-%202%29%20%2B%208)
![\sf \: multiply : \\ \tt - 24 - 8 + 24 + 8](https://tex.z-dn.net/?f=%20%5Csf%20%5C%3A%20multiply%20%3A%20%20%5C%5C%20%5Ctt%20%20-%2024%20-%208%20%2B%2024%20%2B%208)
![\sf simplify : \\ \sf- 32 + 32 \\ \sf0](https://tex.z-dn.net/?f=%20%5Csf%20simplify%20%3A%20%20%5C%5C%20%20%20%5Csf-%2032%20%2B%2032%20%5C%5C%20%20%5Csf0)
therefore
we have proven that x+2 is a factor of 3x³-2x²-12x+8
let's factorise 3x³-2x²-12x+8
- factor out x²: x²(3x-2)-12x+8
- factor out -4: x²(3x-2)-4(3x-2)
- group: (x²-4)(3x-2)
- use a²-b² = (a+b)(a-b):
and
we are done
Answer:
p=7/2
Step-by-step explanation:
Rember that a p series can be represented by
![\frac{1}{k {}^{p} }](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bk%20%7B%7D%5E%7Bp%7D%20%7D%20)
Here, notice that
![\frac{1}{8 \sqrt{2} } = \frac{1}{2 {}^{3} \sqrt{2} } = \frac{1}{2 {}^{3} \times 2 {}^{ \frac{1}{2} } } = \frac{1}{2 {}^{ \frac{7}{2} } }](https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7B8%20%5Csqrt%7B2%7D%20%7D%20%20%3D%20%20%5Cfrac%7B1%7D%7B2%20%7B%7D%5E%7B3%7D%20%5Csqrt%7B2%7D%20%20%7D%20%20%3D%20%20%5Cfrac%7B1%7D%7B2%20%7B%7D%5E%7B3%7D%20%20%5Ctimes%202%20%7B%7D%5E%7B%20%5Cfrac%7B1%7D%7B2%7D%20%7D%20%7D%20%20%3D%20%20%5Cfrac%7B1%7D%7B2%20%7B%7D%5E%7B%20%5Cfrac%7B7%7D%7B2%7D%20%7D%20%20%7D%20)
This is true for all parts of the series because
![27 \sqrt{3} = 3 {}^{ \frac{7}{2} }](https://tex.z-dn.net/?f=27%20%5Csqrt%7B3%7D%20%20%3D%203%20%7B%7D%5E%7B%20%5Cfrac%7B7%7D%7B2%7D%20%7D%20)
So p=7/2
In order to solve this problem, you need to use a geometric series:
![S_{n} = \frac{ a_{1}(1 - r^{n}) }{1 - r}](https://tex.z-dn.net/?f=%20S_%7Bn%7D%20%3D%20%20%5Cfrac%7B%20a_%7B1%7D%281%20-%20r%5E%7Bn%7D%29%20%7D%7B1%20-%20r%7D%20%20%20)
where:
a₁ = first term of the series = 36000
r = common rate = 10% raise, therefore 1.10
n = number of terms = 5
Therefore,
<span>
![S_{n} = \frac{ 36000(1 - 1.10^{5}) }{1 - 1.10}](https://tex.z-dn.net/?f=%20S_%7Bn%7D%20%3D%20%20%5Cfrac%7B%2036000%281%20-%201.10%5E%7B5%7D%29%20%7D%7B1%20-%201.10%7D%20%20%20)
= 219783.60 $
Luke's total earnings in five years are
<span>
219783.60 $.</span>
</span>
Well if it was increased from last year, then it must be dealing with devision to find out the price of last year...
So when you divide the number 287.50 by 15% You get to the total of:
<span>1 916.66667
</span>So your answer is: <em>$1,916.67 </em>