North

The second part is asking "what more specific information do we need about the roots in order to know for certain if f(x) divides g(x) evenly?" The reason why you need to know the multiplicity (the number of times the roots shows up in the factorization) is because a polinomial could be something like (x+10)³ which has -10 as a zero and another polynomial could be (x+10)(x+17). it also has a zero at 10, but (x+10)³ obviously does not go into the other polynomial. This is not a proof because it is not general, but i hope it can further put you on the right track. Sorry for not getting back sooner, hope this is still useful!

For the first part, if one or more of the roots of f(x) has a multiplicity of 2 or greater and g(x) has no roots greater than multiplicity 1. an example polynomial you could use for f(x) is:

f(x) = (x+3)(x+3)(x-4)(x-8) 

and for g(x):

g(x) = (x+5)(x+3)(x-2)(x-4)(x-8)

I'll let you expand those if you need to.

For the second part, f(x) evenly divides g(x) if all the overlapping roots have the same multiplicity. I'm not too good at writing out proofs though, so you can try to prove my statement yourself or wait for someone else to. 

Hope I was of some help!

You are right, I made a mistake. Sorry!

Factor the denominator to get

(x+1)/[(x+2)(x-2)]

and then check what happens around all the critical points (-2, -1, 2) to see that the solution is (in interval notation):

(-2, -1) U (2, ∞)

Hope this solution helps!

it seems like you are missing some numbers in the question. If you reply to this with the complete form of the question I will answer it for you in a bit. Your question seems similar to this one though:

to solve the question, you need to find all the multiples of 16 and 25, and then subtract the overlap.

4000/16 = 250, so there are 250 multiples of 16 that work.

4000/25 = 160, so there are 160 multiples of 25 that work.

The overlap in this question occurs when a number is divisble by both 16 and 25. We counted those overlapping numbers twice so we need to go back and subract them off once.

4000/(16*25) = 4000/400 = 10, so there are 10 overlapping numbers.

The final answer to this problem, therefore, is 250 + 160 - 10 = 400

Hope this solutions helps!

An x between 0 and 360 which satisfies the given sine and cosine is 

x = 150˚

Now we need to find 2520/360, which is 7. 

All that is left to do is add 2520 to 150, which is 2670˚.

I hope this solution helps!

You can just brute force this one:

6 cases:

• all 6 blue
  • this case is pretty easy to see, (6,0). 1 solution for this case.
• 5 blue 1 green
  • two people sit next to the person with green eyes and everyone is sitting next to someone with blue eyes (6, 2). 1 solution for this case as well.
• 4 blue 2 green 
  • if the greens are next to each other, there is one solution (6, 2)
  • if the greens are one apart, there is another solution (6, 3)
  • if the greens are two apart, there is one more (6, 4), for 3 solutions in this case. 
• 3 blue 3 green
  • all 3 greens next to eachother (5, 5)
  • 1 green separate from the other 2 (5,5)
  • blue and greens alternate (3,3). In this case we have a repeating solution so only 2 more posibilities. 
• 2 blue 4 green
  • same as the 4 blue 2 green, just flipped for 3 more solutions.
• 1 blue 5 green
  • same as the 5 blue 1 green, just flipped for 1 additional ordered pair.
• all 6 green
  • same as all 6 blue, just flipped for 1 additional ordered pair.

Adding up the bolded numbers, we get a final answer of 12 ordered pairs.

Hope this solutions helps!

the equation of the circle you described is

(x-5)²+(y-15)²=130

we are also told that

x = 2y

now all we do is substitute in 2y for x and solve:

(2y-5)²+(y-15)²=130

4y²-20y+25+y²-30y+225=130

y²-10y+50=26

y²-10y+24=0

(y-4)(y-6)=0

so y=4 or y=6

but since we are looking for the maximum value of x, and 2y=x, we want the larger y, 6.

Plugging this into 2y=x, we get

2(6)=x

x=12

Hope this solution is helpful!

The slope of -3/2 means that for every 3 you go down in the plane, you go to the right 2. To get from -2 to 4 for x, you need to move right 6, so you need to move down 9 to stay on the line described, which will bring you to the point (4, -2). So,

y = -2

Hope this helps!