CPhill

OK....let's use your  example  and solve for  "b"

So  we have

M  = ( b - c)  / ( b + c)     multiply both sides by  ( b + c)

M ( b + c)  = b - c         distribute the M on the left 

Mb + Mc  =  b  - c      subtract b, and Mc from both sides

Mb - b   =   - Mc -  c       multiply both sides by - 1

b - Mb  = Mc + c        factor both sides

b( 1 - M)  =  c ( M + 1)      divide both sides by (1 - M)

b  =  c ( M + 1) / ( 1 - M)

So......back-substituting, we have

E_max  =  E_min ( M + 1) / (1 - M)

(x^3-5x^2+7x-12)÷(x-4)

             x^2   -  x    +  3

x - 4  [  x^3  -5x^2  + 7x  -  12  ]

            x^3 - 4x^2

          ______________________

                  - x^2  +  7x

                  - x^2  + 4x

                __________________

                            3x   -   12

                            3x   -   12

                          __________

The result is in red

2)  Here's the graph : https://www.desmos.com/calculator/hz0nsgeb2n

The x intercept   is  (1, 0)

The y intercept is (0, - 3)

1)  (a)  There will be as many roots of a polynomial as its degree  [  some may be repeated roots ]

So....the number of roots here    = 2    since this is a second degree polynomial

(b)   2x^2 + 4x + 7  = 0     subtract 7 from both sides

2x^2 + 4x  =  - 7      factor out the 2 on the left

2 ( x^2 + 2x )   =  - 7    divide both sides by 2

x^2 + 2x  =  -7/2  

Take 1/2 of 2  = 1...square it = 1....add it to both sides

x^2 + 2x + 1   =  -7/2 + 1      factor the left side, simplify the right

(x + 1)^2  = -5/2     take both roots

x + 1  =  ±√ [ - 5/2]  =  ±i √ 5 / √2  =   ±i √10 /2

x + 1  = ±i√10 / 2    subtract 1 from both sides

x =  -1 ±i√10 / 2

x =  [  -2  ± i √10 ]  / 2

Thanks, EP....I sorta' expected that THAT was the question that was meant to be asked, but I wasn't sure....

In  the form    ax^n  + bx^m +.......  + c  =  0

The possible rational  roots  [ rational meaning that we can express a number as a fraction]  are all the possible factors  of  "c"   over all the possible factors of "a"

So we have

9x^3  + 14x^2 - x + 18   =  0

So...all the possible  factors of  c = 18  are ±18, ±9, ±6, ±3, ±2, ±1

And all the possible factors of  a  = 9   are  ±9, ±3, ±1

So....in your answer...all the possible rational roots  are    ±1/3  and ±3/1

Note that ±1/2  isn't possible because ±2  isn't a factor of  9

And  ±1/ 18  isn't possibe because ±18  isn't a factor of 9

Note something, PhoenixForever......this does not mean that these are necessarily the roots of the equation....they are just possible roots...!!!

Here's a resource I always like, Purplemath....it explains this in more detail than I can :

https://www.purplemath.com/modules/rtnlroot.htm

Let x  be the length of GA

So.....the perimeter of the adult model is the sum of all the side lengths....so

56 + 56 + 35 +  x   =  177    simplify

147   +  x  =   177      subtract  147 from both sides

x  =  30   =    length of GA

I think EP and Melody  have secret powers.....!!!!

Mmmmm... a picture of nothing....that's an interesting concept  !!!!

Let AX  =  x    and  BX  = 50 - x

Angle C is bisected...so....we have this relationship

CA / CB   =  AX / BX

28/ 56  =  x  /  [ 50 - x ]        

1 / 2   =  x / [ 50 - x]      cross-multiply

50 - x   =  2x      add x to both sides

50  =  3x      divide both sides by 3

50/3   =  x   =  AX