CPhill

What is   "E"   ???

Is this process confusing???....it might not be the same as what you are presented....but....somehow....I find it easier

29)   factor top / bottom

( x + 2) ( x - 2)

___________

-3 ( x + 3)

"Holes"  will only be created when we can "cancel" factors

Here....nothing "cancels"  ....so......no "holes"

We will have  a vertical asymptote at x  = -3    and something known as a "slant asymptote"

The latter will occur because we have a greater degree polynomial divided by a lesser degree polynomial

30)   factor top / bottom

(x + 3) ( x - 2)                    (x + 3) ( x - 2)

______________   =       _____________

3 ( x^2 + 4x + 3)               3(x + 3) ( x + 1)

Note that  the  "x + 3"  factor  will "cancel"

So.....to find the "hole".....set    x +  3  =  0    and solve for x......and we have

x + 3  =   0

x  =  -3

So.....we have a hole at  x  = -3

See the graph, here :  https://www.desmos.com/calculator/oshh1sofob

RP.....If you hold down the left button on your mouse/laptop  and drag the cursor along the curve...note....at  x  = -3.....you will see "undefined"   =  (a "hole" here )

Simplify the top

Cross-multiply   and keep the same sign between the terms as we have between the fractions

5a + 4a^2

Take the product of the denominators

4 * 5  =  20

So.....we have

[ 5a + 4a^2]              25

__________  ÷       ___

       20                      1

Just like "regular" fractions......Invert the second fraction and multiply

[ 5a + 4a^2 ]                  1

___________      x      _____

     20                            25

5a  +  4a^2

_________

   500

MA = 20   and MX  = 9

And the perpendicular bisector  of  AB  =  CM

So, triangle MAX will be a right triangle with angle AMX  = 90°

Draw  XN   perpendicular to AC

And AD bisects angle BAC...so....

And angle MAX = angle NAX

And angle AMX = angle ANX  =  90°

And AX  is  a common side to triangles AMX  and ANX

So....by   AAS, triangle  AMX  is congruent to triangle ANX

So.....MX = NX  =  9   =   the distance from X to AC   

We can use a little logic, here

Let the sides  of the triangle be  AB, BC, AC

We have a 6-8-10  right triangle

The  area  =  (1/2)  [ Product of the legs ]  =  (1/2) [ 8 * 6 ]  = 24

Let the center of the incircle  = O

We can divide the right triangle into  three smaller triangles  =  AOB, BOC and AOC

The inradius will be tangent to each side of triangle ABC, so it will form an altitude to each of the three smaller triangles

So.....the area of ABC  = (1/2) (inradius) [ AB + BC + AC]

24   =  (1/2) (inradius) [ AB + BC + AC ]

24 / (1/2) =  (inradius)  [ perimeter ]

48  =  (inradius)  [ 6 + 8 + 10 ]

48  = (inradius) [ 24 ]

inradius  =  48 / 24   =   2 units

This is a little easier than you might realize

Let us call the sides AB, BC   and AC

And call the center of the incircle  O

So....we can divide the larger triangle into three separate triangles  - AOB, BOC  and AOC

The inradius of the incircle forms an altitude of each triangle since it is perpendicular to each side

So.....the  area, A,  of the larger triangle can be expressed  as

A = (1/2)  [ altitude * ( sum of the bases ) ]  = 

(1/2) [ (1) * perimeter ]  =

 (1/2)  [ (1)(AB + BC + AC ) ]  =

(1/2) [ 1 * 12 ]  =

(1/2) [ 12]  =

6 units^2

Let's do some Algebra, first

Letting x  = 0......the y intercept  is  at     - 3

We  have

1x^3  + 2x^2  - 3

Note that, if we add the ntegers in red and get 0, then 1 is a root  (an x intercept)

So  ( x - 1)  is a factor

To find if we have any other real roots....perform some polynomial division

            x^2  + 3x  + 3

x - 1  [  x^3  + 2x^2  +  0x   -   3  ]

            x^3  -  1x^2

          ______________________

                      3x^2  + 0x

                      3x^2   -  3x

                      ________________

                                 3x  - 3

                                 3x  - 3

                                 ______

The remaining polynomial  is    x^2  + 3x   + 3

The discriminant  is     3^2 - 4 (1) (3)  =  -3

Since this is < 0......we have no more real roots

So.....

The y intercept  is - 3

And the x intercept is  1

Here's the graph : https://www.desmos.com/calculator/kfgpcs3g09

Let   BP  = x       and  CP  = 20 - x

In this situation, the altitude will be a "geometric mean"....such that

x            6

__  =   _____

6         20 - x

Cross-multiply

x (20 - x)  =  36

20x - x^2  = 36      rearrange as

x^2 - 20x + 36  =  0        factor

(x - 2) ( x - 18)  =  0

Set each factor to 0   and solve for  x  and  we get that  x  =  2   or x  = 18

So.....depending upon which  vertex is labeled as B  and which is labeled C.....

BP = 2  and CP  = 18    or

BP = 18  and CP  = 2

Either way.....the hypotenuse is divided into a ratio of 18 : 2

Good....but cold   !!!   LOL!!!