CPhill

1.   P(x)=x^3 + 9x^2 + 24x + 20

By the Rational Roots Theorem, -2 is one root...so we have

-2     1    9      24      20

             -2     -14    -20

----------------------------

        1    7      10      0

The remaining polynomial  is x^2 + 7x + 10    which factors as (x + 2) (x + 5)

So the factorization of this is    P(x)  =  (x +2)(x + 2) (x +5)  = (x + 2)^2 (x + 5)

2.  We have

                            x^2   + 7x   + 10

             --------------------------------------------

x + 2    [  x^3  + 9x^2  + 24x  + 20 ]  

                x^3  + 2x^2

             -------------------------------------------

                           7x^2  + 24x

                            7x^2 + 14x

             --------------------------------------------

                                        10x  + 20

                                         10x + 20

             ------------------------------------------

                                                     0  remainder

So, now notice, tyler, the quotient polynomial,  x^2 + 7x + 10,   factors as before  in the first part

So the three factors  are (x + 2)(x + 2)(x + 5)  =  (x + 2) ^2 (x + 5)  .....as before....

The equation is

(y - 5)^2 / 9 - (x - 4)^2/ 4  = 1

Transforming this, we have

4(y - 5)^2 - 9(x - 4)^2  = 36

4(y^2 -10y + 25) - 9(x^2 - 8x + 16) =36

4y^2 - 40y + 100 - 9x^2 + 72x - 144  = 36

4y^2 - 9x^2 - 40y + 72x - 80 = 0

I think it was long about ...  DCLXVI  ......give or take

Let the son's age now  = x   ....  then the father's age is 4x

And in four years, we have

3(x + 4) = 4x + 4   simplify

3x + 12 = 4x + 4

x = 8   and that's the son's age now

And the father's age now is 4x = 4(8)  = 32

The fraction that are children =

1 - (1/4) - (3/5)  =

20//20 - 5/20 - 12/20  =

3/20

The amount is given by

15000(1.06)^33   .....I let you plug this into the calculator to get the answer.....

The radius - r -  of an octagon with side length s can be figured thusly :

(1/2)s / sin 22.5  = r   =  s/(2sin22.5)

And the total area is gven by

A = 8*(1/2)r^2sin45 .... and substituting for r, we have

A = 4s^2/[2sin22.5]^2 * sin 45

A =  s^2 [sin45/[sin22.5]^2]  

So...if the side length is 7 times as great....the area =

A = (7s)^2  [sin45/[sin22.5]^2] = 49s^2 [sin45/[sin22.5]^2]  = 49 times as large = 49 x 35 = 1715 sq units

Notice that the first row = 4

And the next row has 8

And the next 12, etc.

So...we have

4(1) + 4(2) + 4(3) +..........+ 4(55)  =

4 [ 1 + 2 + 3+......+ 55]

And the sum in the brackets  is equal to ....    (55)(56)/2 = 1540

So,  4 times this =  6160

I always think of things like that in this way :

(20 + 5) (20 + 5)

I see that multiplying the first two terms gives me 400

Then, I have 5 x 20,  twice  = 200

Finally, the last two terms multiplied together = 25

So..adding the results, we have......400 + 200 + 25 = 625

{However....since I just did this on "paper".....I guess it wasn't done in my head after all....!!! }