Expert: Scott A Wilson - 12/17/2009

Question
I am having trouble finding zeros.  Find the zeros of 5x3-17x2-134x+56.

Answer
X   Y
0.00   56.00
1.00   -90.00
0.38   2.38
0.40   0.08
0.40   0.00

I did that using the Secant Method.
I quickly found one of the roots to be 2/5 = 0.4.

Next, we do synthetic division with (x-0.4).
That gives us the other factor is 5x²-15x-140.
Checking, 140*0.4 is 56, so that is right.

Note that to be in integers, (x-0.4) is the same as (5x-2)/5.
If the 5x²-15x-140 is divided by 5, the result there is x²-3x-28.
This means the original can be factored into (5x-2)(x²-3x-28).

Using the quadratic formula, we have a=1, b=-3, and c=-28.  That gives us (3±√(9+112))/2.
This tells me there was no need to do that, since after looking at the problem,
I can see that it is (5x-2)(x-7)(x+4).

If I had found x=-4 or x=7 at first, it might have gone differently,
but the result would be the same.

From that equation, since we are fnding where 0 = 5x³ - 17x² - 134x + 56,
it can be seen that that is the same as 0 = (5x-2)(x-7)(x+4).

From here, all that is left to do is to solve 5x-2=0, x-7=0, or x+4=0 to get 3 solutions.
To verify that these are correct (or verify that arithmatic can be done),
try putting these answer back into the original equation.