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Part 1:

If  f(x) = 0  when  x  =  -3, 4, or 8 ,   then  f(x)  in its factored form is...

f(x)  =  (x + 3)^a(x - 4)^b(x - 8)^c       where  a, b, and c are positive integers.

If  g(x) = 0  when  x  =  -5, -3, 2, 4, or 8 ,   then  g(x)  in its factored form is...

g(x)  =  (x + 5)^d(x + 3)^e(x - 2)^f(x - 4)^g(x - 8)^h       where  d, e, f, g, and h are positive integers.

So for example,  g(x) / f(x)  could be...

\(\frac{g(x)}{f(x)}=\frac{(x+5)(x+3)(x-2)(x-4)(x-8)}{(x+3)(x-4)(x-8)}\)            which reduces to....

\(\frac{g(x)}{f(x)}=(x+5)(x-2)\)

In this case,  g(x)  is divisible by  f(x)  with no remainder.

However,  g(x) / f(x)  could also be...

\(\frac{g(x)}{f(x)}=\frac{(x+5)(x+3)(x-2)(x-4)(x-8)}{(x+3)^2(x-4)(x-8)}\)            which reduces to...

\(\frac{g(x)}{f(x)}=\frac{(x+5)(x-2)}{(x+3)}\)

In this case,  g(x)  is not divisble by  f(x)  with no remainder.

So it is not necessarily true that  g(x)  is divisible by  f(x)  with no remainder.

An example is   f(x)  =  (x + 3)²(x - 4)(x - 8)   and   g(x)  =  (x + 5)(x + 3)(x - 2)( x - 4)(x - 8)

Part 2:

For  g(x)  to be divisible by  f(x) ,  f(x)  has to have the same zeros as  g(x)  and each zero of  f(x)  has to have a multiplicity less than or equal to that of g(x) .

Connecting the centers  of the smaller circles  forms a hexagon  with  sides  of  6  units

But segments drawn from the center of the larger circle to each of these  6 centers  will form 6 equilateral triangles......so....a side of one of these triangles =  the side of the hexagon  =  6 units  (1)

And the distance from  a smaller circle center to the edge of the larger circle  = the radius of a smaller circle   = 3 units   (2)

So....the radius of the larger circle  =  (1)  + (2)  =    6 + 3  =   9 units

So...its diameter  =  2 * 9    =    18 units

but Guest you got to think without math human civilization would never have all the stuff it has now like cell phones or laptops or TV

Math isn't my strongest subject, though there are somethings that I enjoy doing in math. 

Yes i have trouble comprehending it too DarkCalculus.  

How can anyone not like maths LOL

I see you beat me to the finish line MIRB16.      

At least we got the same answer, always a good sign :)     

A line segment is broken at two random points along its length. What is the probability that the three new segments can be arranged to form a triangle?

I would do this with a boolean contour map.

The line segment can be any length so let it be 1

Let the segment by cut into segments that are    x,  1-y and   y-x    units long.

(If you add these you can see that the total length is 1 unit.)

No each of these lengths must be between 0 and 1 unit long.

so

\(0 x \quad (3) \qquad and \qquad y

I need to plot these to get the full sample space.  The dark triangle is the sample space.

Now in order for the 3 peices to from a triangle, the sum of the two little ones must be greater than the length of the long one.

Remembering that the  segments  are    x,  1-y and   y-x    units long.

\(x+(1-y)>y-x\\ 2x+1>2y\\ y

AND

\(x+(y-x)>1-y\\ y>1-y\\ 2y>1\\ y>0.5 \qquad(6)\)

AND

\((1-y)+(y-x)>x\\ 1-x>x\\ 1>2x\\ x<0.5 \qquad (7)\)

So a quarter of the sample space is the area of the desired region.

P(the three pieces form a triangle) = 0.25

V_cube   =  side^3

So

698  =  side^3     take the cube root of each side

³√698   =  side  ≈   8.9  cm

y   =  -2*3^x

The graph of the function passes through the point (0,−2)

The range of the function is the set of all real numbers less than 0.

The domain of the function is all real numbers.

Very nice, MIRB16  !!!

thanks

You are loved.

you don't like math?????

hey, guys, I feel a lot better now (but I still have a massive headache) but thank you ilovepuppies1880 and CPhill for trying to cheer me up

No, not really

yes I love math otherwise I would not be here

Suppose the points where we break the line are x and y, where 0 < x < y < 1. So our "possible" region is the following triangle on the plane:



The three sides have lengths x, y-x, and 1-y. We must satisfy the Triangle Inequality on all three sides, meaning that the sum of any two of these lengths must be greater than the third. This gives us

1.  : This gives y>1-y, so y> 1/2.

2. : This gives 1-x> x, so x<1/2.

3.  : This gives  , so  .

Adding these lines to the diagram gives us the darker "success" region.



The darker region makes up 1/4 of the total region, so the probability is 1/4.

Thank you!!!!!!!

1)  I don't really know much about "playing cards", but isn't it a simple matter of counting?

There are four "4s" in a deck of 52 cards, isn't that right? So, the probability is:

4/52. But the first 4 could a 4 of club(whatever that means!!), so you have only 12 cards of club left.

So, the probability of any of the 12 cards out of 51 remaining cards would be:

12/51. But this 2nd draw could be a 2 of club, so would have left three "2s" out of the remaining 50 cards, 3/50. So, the overall probability would be(if I understand) "playing cards" :

4/52 x 12/51 x 3/50 =144/132,600 =6 / 5,525

Note: Somebody who understands this much better than I should check this out!.

2) The probability of rolling ANY number on 1 die is 1/6. But, you have 5 dice, so the probability of rolling any particular number would be: 5 x 1/6 =5/6. And the probability of rolling the same number again would still be: 5 x 1/6 =5/6. The remaining 3 dice could any number or 6//6 x 6/6 x 6/6. So, the probability of getting two of a kind would be:5^2 x 6^3 / 6^5 =25 / 36

Note: Somebody should check this one as well!.

3) Melody is pretty good at these seating arrangements, so she should take a look at this one!

The directrix  is 8  units below the focus

So...the vertex  is  ( 2, (6 + -2)/2 )  =  (2, 4/2) =  (2, 2)

And  p  is the distance between the vertex  and directrix  =  [2 - (-2)]  =  4

So....we have the form

4p(y - k)  = (x  - h)^2       where  (h, k) is the vertex  and p  = 4

So  we have

4(4)(y - 2)  = (x - 2)^2

16 (y - 2)  = (x - 2)^2

(y - 2)  = (1/16)(x - 2)^2

y  =  (1/16)(x - 2) + 2

Angle  C  = Angle R

And  angle  C  =   180  - 45  - 60   =  75°  =  angle  R

The area  of  the triangle  is

(1/2) (3)^2 √3/2  =  (9/4)√3  units^2

The  area  that is no more than  unit from A  is (1/6) of a circle with a center at A and a radius of 1 =

pi *(1/6)(1)^2   =   pi / 6  units' 2

So....the probability  that X  is no more than 1 unit from A  is

[pi / 6 ]   /  [(9/4)√3 ]  =

4 pi  / [ 54√3]  =

2pi / [27√3 ]  ≈  0.134  ≈   13.4 %

8)The center of a regular octahedron is 1 foot away from every vertex. What is the volume of the octahedron in cubic feet?

The octahedron will form two congruent pyramids

The edge of  one of these  is   2/√2  = √2

So....the area of the base of one of the pyramids is (√2)^2  =  2 ft^2

And the height  = 1 ft

So.....the volume  of one of the pyramids  is

(1/3) (area of base)  ( height)  =

(1/3) (2) (1)  =  2/3  ft^3

And by symmetry we have    

4/3  ft^3

Thanks so much CPhill, turn's out #3 was 18 but that might have been a glitch or a typo.

Thank you!