🚀 go-pugleaf

Do you know how to do synthetic division?  I will try to show
you how to set it up here-

1) You take the coefficients (A,B,C,and D) and write them in a
line-  like so-

	A  B  C  D

2) Then you look for the bounds (both upper and lower) by doing
the following- Take the coefficient of the cubed term (a) and
make a list of ALL of its factors.  Then do the same with the
constant term (d).  Then you divide each factor of A by each
factor of B.  This will give you the list of possible RATIONAL
roots.  Then you pick one , and put it in a box beside you
coefficients in the synthetic division setup-

  F| A B C D
  -|_________
Do
Then you bring A down below the bottom line, and multiply it by
F (the factor) that value (call it X) then goes below B. You
add X to B, and place it below the li under B . Multiply then
add,  until you have gone all the way thru D.

OK, now when the sums at the bottom alternate as positive and
negative, it is the lower bound.  When they are all positive,
it is the upper bound.  Once you have both, you take all the
values of the possible rational roots that are between the
upper and lower bound, and use synthetic division with them.
If the final sum at the end of the problem is 0, then the
number is a root.  Then you take the remaining sums at the
bottom and use them as coefficies for a quadratic.

Now, an example:   3     2   
		 4x +  2x + x +8

First, I am not sure of the roots of this equation, so you will
see me do my work here.

Factorsof A: +/-1,+/-2,+/-4
Factros of D: +/-1,+/-2,+/-4,+/-8

Possible rational roots: 1, -1, 2, -2, 4, -4, 1/2, -1/2, 1/4,
-1/4

Synthetic division:

1 | 4 2 1 8
      4 6 7
  _________
    4 6	7 15 <-- since these are all positive, all rational
roots are below 1.

-4 | 4   2   1   8
       -16  56   -
     ______________
     4 -14 57 - <-- This will obviosly be neg, so -4 is a lower
bound.

Ok, I'm running out of time, but if you can see what I am doing
here, you should be able to figure it out.  Good Luck!

P.S.- consult a textbook or a math teacher for a more in-depth
explanation of synthetic division.