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If 1 is a root...let's perform  some synthetic division

1  [  1     a + 17    38   - a      - 56  ]

                     1      a + 18        56

   _________________________

      1     a + 18        56            0

The remaining polynomial is    

x^2  + (a + 18)x  +  56

Factors of 56  are  1   2   4    7   8   14   28   56

Since a  is positive...only the pairs  2, 28    and  1, 56   will give us what we want

So...the possibiliites for the factorization of this are

(x +28) ( x + 2) =  x^2  + 30x  + 56       so  a  =  12       or

( x + 56) ( x + 1)  = x^2  + 57x  +  56    so   a  = 39

Checking this

x^3 + 29x^2 + 26x  - 56  = 0  produces the three roots of 1, -28  and  -2

And

x^3  + 56x^2 -1x - 56  = 0   produces the three roots   1, -1, - 56

I'm assuming  that  Contractor 1 fees are modeled by

y = 200 + 50x

And we know that Contractor 2  fees are modeled by

y = 150  + 55x

(a)  Contractor 2 charges  $5 more per hour than Contractor  1

(c) The flat fee for Contractor 1  is $50 more than that for  Contractor  2

oh, there i edited it :)

No prob, skye  !!!!

Don't see any  formulas. hoiuu  (???)

thank you

Yep...I can see it  now  !!!

And it is correct........

1. The graph on the top right is correct...it crosses the x axis   at  - 6 , -2,, 1 and 3

2.  The graph at the bottom left is correct ...it crosses the x axis at  -4, 0 and 2

Does it show up now?

A)  Your graph doesn't show up, hoiuu...but here it is : https://www.desmos.com/calculator/x359nx2bzz

The x axis are the hours worked and the y axis represents the total cost   

B)  Correct!!!.....  $200 flat fee for doing the job  plus $50 per hour

C) All correct   !!!

Good job  !!!!!:

Thanks, DC  !!!

Here's the way  DC arrived at  the answer

He drank   [ 275 - 210 ]  = 65 mL

To find the rate he drinks per second :      65   / 13    =  5 mL  per second

To find out how long it took him to drink all the slushy.....   275  / 5      =    55  seconds to drink it all

And   55 - 13  = 42 seconds to drink the remaining amount

2𝑥4 + 14𝑥3 + 6𝑥2 + 14𝑥 + 2 = 0

This has an irreducible factorization.....there is an algebraic method for solving a quartic...but....it's EXTREMELY messy  !!!

We  can give approximate solutions with a  graph :   https://www.desmos.com/calculator/o4ek4yiqx4

There are only two real  roots  x ≈ -6.705   ans  x ≈ -0.149

The non-real roots are   x ≈ -0.07295 + 0.99734 i     and   x ≈ -0.07295 - 0.99734 i

[ The non-real roots were calculated using  Wolfram Alpha  ] :  https://www.wolframalpha.com/input/?i=2x%5E4+%2B+14x%5E3+%2B+6x%5E2+%2B+14x+%2B+2++%3D+0

The last account I used before I left for awhile was RosyWinterCat

 5x^3+8x^2−7x−6

If   (x + 2)  is a factor then  -2   must be a root

So...putting  -2  into   5x^3+8x^2−7x−6  and evaluating we get

5(-2)^3  + 8(-2)^2   - 7(-2)  - 6   =

-40  +  32     +  14   - 6    =

-8    +  8    =   0

We get a 0....this tells us that -2 is a root and ( x - -2)  = (x + 2)  is a factor

If  x + 1 is a factor....then -1 must be a root

So

5(-1)^3  + 8(-1)^2   - 7(-1)  - 6  =

-5    +  8   +   7    - 6  =

3   +  1    =

4

This does not produce a  0...so......-1 isn't a root and  ( x - - 1)   = ( x + 1)  isn't a factor  !!!

Kool!

Lol hey! Yeah twins haha 

HAHAHA!!!!....you can't  beat two "Roses".....!!!!!

We have the form   y  = mx + b

Where  b  is the y intercept and  m is the slope of the line

The points  (0,2)  and (3,0)  are on the graph

The slope of the line is   [  0 - 2 ] / [ 3 - 0 ]   = -2/3   =  "m"

The y intercept  is  2

The only  equation with a slope  of  -2/3  and a y intercept of 2  is   "H"   ⇒   y  = (-2/3)x + 2

Hey CPhill, I've missed y'all!

Another Rose?!? I just got here but that is really cool twinning and nice to meet you

Hey, Rose....good to see you back  !!!!

1) f(-3)   = 36

2)  4x^3+9x^2−x−6

-1 is  a root because    4(-1)^3  + 9(-1)^2 - (-1)  - 6  =     -4   + 9 + 1   - 6 =  -5 + 5  = 0

So  (x + 1)  is one factor

Performing some synthetic division  to find the remaining polynomial  we have

-1  [  4   9     - 1   - 6  ]

            -4      -5     6

     ________________

       4    5      -6

So....the remaining polynomial is    4x^2  + 5x  - 6

Factoring this we have

( 4x - 3) ( x + 2)

So...the factors are  ( x + 1) (x + 2) (4x - 3)

3) Since we know that 3 is a zero...we can perform some synthetic division to find the other factors

  3   [ 1  - 4  - 3   18  ]

               3   -3   -18

      _______________

         1   -1   -6

The remaining polynomial is    x^2  - x   - 6    which factors as    (x - 3) ( x + 2)

So...the factorization is

(x - 3) ( x - 3) ( x + 2)   =  (x + 2)  (x - 3)^2

4) Possible  rational roots   =  all the factors of 10  =  1, 2, 5, 10   over all the  factors of 4  =  1, 2, 4

So...   ±5/2  , ±1/2  and ± 2

the answer to your first question is that he drank it at 5 milliliters per sec. 

and the answer to your final question it will take him 42 seconds to drink all the slushy and a big brain freeze 

Or look at it as 100% of 180 + 10% of 180 + 5% of 180

100% of 180 = 180

10% of 180 = 18

5% of 180 = Half of 10% of 180 = 18/2 = 9

180+18+9 = 207

It is not as tough as it looks! It is just a lot of work! Where in the world did get this from? At any rate, you can calculate the individual permutations of each sum using this complicated-looking formula and then sum them all up:
S=28 -35; N=8; sum(k, 0, ((S - N)/6), ((-1)^k * (N nCr k) * (S - 6*k - 1) nCr (N - 1))

However, fortunately, there is an elegant shortcut to doing all the above work, thanks to Wolfram/Alpha!!.
Simply Expand this polynom: [n + n^2 + n^3 + n^4 + n^5 + n^6]^8 and then just add up all the coefficients of n^28 to n^35 inclusive, because the formula above calculates exactly the same coefficients individually. Here is the expansion of the above polynom:

n^48 + 8 n^47 + 36 n^46 + 120 n^45 + 330 n^44 + 792 n^43 + 1708 n^42 + 3368 n^41 + 6147 n^40 + 10480 n^39 + 16808 n^38 + 25488 n^37 + 36688 n^36 + 50288 n^35 + 65808 n^34 + 82384 n^33 + 98813 n^32 + 113688 n^31 + 125588 n^30 + 133288 n^29 + 135954 n^28 + 133288 n^27 + 125588 n^26 + 113688 n^25 + 98813 n^24 + 82384 n^23 + 65808 n^22 + 50288 n^21 + 36688 n^20 + 25488 n^19 + 16808 n^18 + 10480 n^17 + 6147 n^16 + 3368 n^15 + 1708 n^14 + 792 n^13 + 330 n^12 + 120 n^11 + 36 n^10 + 8 n^9 + n^8.  
Now Just add up the coefficients: =135,954+133,288+125,588+113,688+98,813+82,384+65,808+50,288 =805,811 - total permutations of all the 8 tosses of the dice!!
Therefore, the overall probability is: 805,811 / 6^8 =47.9759%. And that is the END!.