# Thread: Zeros and Synthetic Division

1. ## Zeros and Synthetic Division

Solve:

x^3+3x^2-5x-15=0

(a)List all potential zeros

(b)Do synthetic division to solve then finsh

Thanks

2. Lets look at the very similar equation:

$\displaystyle x^3 + 2.x^2 - 7·x - 14=0$

First thing to do is sketch the graph of:

$\displaystyle x^3 + 2.x^2 - 7·x - 14$

to see if we can find the approximate location of its zeros.

From the sketch in the attachment we see that this has zeros
near -2.7, -2, and +2.7. Trying these out we find that -2 is
an exact root of the cubic. So we may write:

$\displaystyle x^3 + 2.x^2 - 7·x - 14\ =\ (x+2)(A.x^2+B.x+C)$
Now you can use synthetic division to find the quadratic, which
in this case turns out to be:

$\displaystyle x^2-7$
which may be factorised either by inspection of by using the
quadratic formula to find its roots, and hence its linear factors.

So:
$\displaystyle x^3 + 2.x^2 - 7·x - 14\ =\ (x+2)(x+\sqrt 7)(x-\sqrt 7)$
So the solutions of the cubic are $\displaystyle -2,\ -\sqrt 7, +\sqrt 7$.

Your problem can be solved in a similar manner.

RonL