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Using synthetic division, we have

-1  [  2    0     0        k           1     ]                              

             - 2    2    -  2      - k + 2                                 

        ____________________                      

        2    -2     2    k - 2   - k + 3 

1  [  3    0      k         0        2        ]

             3      3       k + 3   k + 3

      _____________________

       3     3   k + 3    k + 3    k + 5   

Equate remainders

- k  + 3   =  k +  5

-2k  =  2

-1  =   k

3)             Fake gold bricks are made by covering concrete cubes with gold paint, so the cost of the paint is proportional to their surface area while the cost of the concrete is proportional to their volume. If a 1 inch cube costs $1.30 to make while a 2 inch cube costs $6.80, then how much would a 3 inch cube cost?

Let the cost of paint per  square inch  = P

Let the cost of concrete per cubic inch = C

Surface area of 1 inch cube =  6  in^2     and  of a 2 inch cube  = 24 in^2 

Volume of 1 inch cube =  1 in^3     and of a 2 inch cube  =  8 in^3

So...we have the following system

6P + 1C  = 1.30       ⇒   -24P - 4C  = - 5.20     (1)

24P + 8C  = 6.80        (2)

Add (1)  to (2)   and we have

4C  =  1.60      divide both sides by 4

C =  .40

And     

6P  + 1(.40)  = 1.30

6P  =  .90       divide both sides by  6

P  = .15

And   the surface area of a 3 inch cube  = 54 in^2

And the volume is    27 in^3

So.....the cost of a 3 inch cube is

54 (.15)  + 27 (.40)   =  $ 18.90

Let the time that took you to drive at 60 mph=T

Distance = Time x Speed

60T = 50[T + 2/3], Solve for T 

T = 3 1/3 hours.

Distance =3 1/3 x 60 mph=200 miles you drove.

2)         Anabel's horse runs 24 miles per hour. Jackie's horse runs 28 miles per hour. Both horses stand at the starting line of a straight 2-mile track, and Jackie's horse starts running 30 seconds after Anabel's. How many miles from the starting line does Jackie's horse catch up to Anabel's?

Note.....the number of   30 second intervals in one hour = 120

At 24 mph, Anabel's horse runs    24 / 120  = 1/5 mile  in 30 seconds

So...her horse has a 1/5 mile head start   

And Anabel]s horse runs    24/60  =  2/5 mile in a minute  = 6/15 miles per minute

And Jackie's horse runs  28/60  =  7/15 mile in a minute

So....Jackie's horse closes the distance between  them at 1/15 miles per minute

So......the time it take's Jackie's horse to catch Anabel's  is

( 7/15 - 6/15) M  = 1/5

1/15 M  =  1/5        multiply both sides by 15

M  = 3 minutes......and in 3 minutes, Jackie's horse runs  3 (7/15)  =  7/5 miles

1)        I drove to my parents' house at 60 miles per hour. If I had driven 50 miles an hour instead, then I would have gotten there 40 minutes later. How many miles did I drive?

Distance  / Rate   = Time

Let the time be in hours...40 minutes = 2/3 hrs.......so, equating times

D  / 50       =   D / 60  + 2/3

D / 50  =   [3D + 120 ]  /  180       cross-multiply

180 D  =  50  [ 3D  +  120 ]    simplify

180 D  =  150 D  +  6000      subtract   150 D  from both sides

30D   =  6000      divide both sides by 30

D  =  200 miles    

Note :  " 3 less than  15 "   translates to  15 - 3   ......so..... 

15  -  n

The first answer is correct, NSS

2/3  -  U  =  1/6    where U is the fractional part used

Rearrange as

2/3  - 1/6  = U

4/6  -  1/6  =  U

3/6   =  U

1/2  =  U   ⇒     1/2 of the package used

Can confirm, it is a real place.

This is rather simplistic  - but - since "c" is a constant......  and the graph of a constant is a horizontal line.....the rate of change between any two x values = 0

√[ 9/16 ]     =     √9 / √16     =     3/4

Thanks man, you're the best!

Let  x be the number  of  Ragnarok and  y  the number of Sicarious

Here are the constraints :  

5x + 6y  ≤  240

3x  + 2y  ≤ 120

5x + 18y ≤  540

And we want to maximize this

45x +  20y

Look at the graph , here :

https://www.desmos.com/calculator/imsgkj9fb3

The profit will be maximized at an integer corner point in the feasible region....there is only one at (18 , 25)

So......we should produce 18 of the  Ragnarok  model  and 25 of the Sicarious model

And the max profit is  $ 1310

EDITED  ANSWER CORRECTING AN EARLIER MISTAKE  !!!

f (x) = x^4 + 3x -9

f ' (x)  = 4x^3  +  3

∫  x^4 + 3x - 9  dx   =    x^5 / 5   +  3x^2 / 2  -   9x

Oh my god that is so helpful. Thank you!!!!

We have factored it as two binomials....

However....I believe the book is focusing on the factors  being polynomials with integer coefficients

So

16  - 2x^2   is factored as

2 ( 8 - x^2 )    

They also appear to want the polynomial inside the parentheses to be written in order of descending powers....so....this gives

-2 ( x^2  - 8)

Does that help  ???

I understand how to solve it, I don't understand how to find any result the book finds. Wolfram never really finds the answer my book wants cause it goes WAAAY too deep into solving.

OBT -  The general rule is to first take out any GCF

So we have

16 - 2x^2

2 [ 8 - x^2 ] which factors as

2  [ √8  -  x ]  [ √8 + x ]        note  √8   = 2√2

2 [ 2√2 - x ]   [ 2√2 + x ]

Note that WolftamAlpha  tends to support the same result as one of its answers :

https://www.wolframalpha.com/input/?i=16+-+2x%5E2

BTW.........the "correct"  factorization  is sometimes ambiguous.....!!!!!

Your answer is just as "correct"  as the book's......

https://www.wolframalpha.com/input/?i=(4%2B+sqrt+(2)x+)+(4+-+sqrt+(2)x)

Solve for y:
y^2 + 10 y - 4 = 0

Add 4 to both sides:
y^2 + 10 y = 4

Add 25 to both sides:
y^2 + 10 y + 25 = 29

Write the left hand side as a square:
(y + 5)^2 = 29

Take the square root of both sides:
y + 5 = sqrt(29) or y + 5 = -sqrt(29)

Subtract 5 from both sides:
y = sqrt(29) - 5 or y + 5 = -sqrt(29)

Subtract 5 from both sides:
y = sqrt(29) - 5                    or                    y = -5 - sqrt(29)

I guess nobody knows, huh? *sigh* Looks like I'll just have to hope my answers are what the book wants.

You must have input it wrong, the calculator comes up with no solution.

 x^4 + 9x^3 + 15x^2 + 9x + 14

Split  15x^2  into  14x^2  +  x^2

 x^4 + 9x^3 + 14x^2 + x^2 + 9x + 14     factor as

x^2  [x^2 + 9x + 14 ]  +  1 [ x^2 + 9x + 14 ]   take out GCF

[ x^2 + 1 ] [ x^2 + 9x + 14 ]

[ x^2 + 1 ] [  x + 7 ] [ x + 2 ]

What I want to know is how to decide if the answer I found is the answer the book wants. My teacher counts it wrong even if my solution is correct. 

The book is correct. 16-2x²/2 does equal x² - 8. What you found equates to the exact same thing. My solution: Both of you are right.

Area of one dollar bill  =  100.75 cm^2

Area of 1 billion of them  =  100750000000 cm^2 =  1.0075 x 10^11    (1)

Number of sq cm to 1 sq mi  ≈  2.59 x 10^10  cm^2      (2)

So.....the answer is given by  (1) / (2)   ≈  3.9  sq mi