Expert: Paul Klarreich - 9/27/2008

Question
Hi
How do I factor:8 t^3 - 9 t^2 - 2
I remember learning somethig about dividing the first coefficient and the last one....I cant remember exactly.

Thank you!

Answer
Questioner:   Natasha
Category:  Calculus
Private:  No
 
Subject:  Factoring 3rd degree polynomials with coefficients
Question:  Hi
How do I factor:8 t^3 - 9 t^2 - 2
I remember learning something about dividing the first coefficient and the last one....I cant remember exactly.

Thank you!
.........................................
Hi, Natasha,

Here is what you learned:

If you think  t = a/b is a root of the equation, divide by  (t - a/b) or by (bt - a).  [This is the FACTOR THEOREM.]

If a/b is a root, then 'a' must divide the last term,  2, and 'b' must divide the first term, 8.  [This is the RATIONAL ROOT THEOREM.]

Now apply:

a divides 2, so  a = +-1,  +-2.

b divides 8, so b = 1,2,4,8.

So the possible roots are  +-( 1, 2, 1/2,1/4, 1/8)

I like to use synthetic division:

8   -9    0    -2  (2)
   16   14    28
----------------------
8    7   14    26, no good

8   -9   0    -2  (1/2)
    4  -5/2
------------------------
8   -5  forget it


8   -9   0    -2  (1/8)
    1  -1   -1/8, bad news!
-------------------------
8   -8  -1
etc.  You will find that none of these works.  So there is no rational root and no rational factor of the form  
(at +- b).  You could try quadratic factors, but if you had one of those you would also have a linear factor.

THERE IS NO WAY TO FACTOR THIS POLYNOMIAL.