Question 1034287

Rational Roots Theorem with synthetic division indicate possible zeros to check are the positive and negative
of 1,2,4,5,8,10,25,50,100.  A graphing tool might help to decide which of the possibilities to try first.

Here are some of the productive synthetic-division root checkings.  The first dividend of coefficients to start is
<pre> 1  12  43  22  -138  -280  -200</pre>; the roots being checked are shown to the left, in the "divisor" location.


<pre>
 -2 |    1    12    43    22    -138    -280   -200
    |
    |          -2   -20   -46     48     180    200
    |____________________________________________________
         1    10    23    -24     -90    -100    0
</pre>

<pre>
 -5 |    1    10    23    -24    -90    -100
    |
    |         -5    -25    10    70      100
    |____________________________________________________
        1     5    -2     -14    -20      0
</pre>

<pre>
 -5 |     1     5    -2     -14    -20   
    |
    |          -5    0       10     20
    |____________________________________
          1    0      -2     -4     0
</pre>

<pre>
 2  |     1    0      -2     -4
    |
    |          2      4       4
    |______________________________
          1    2      2      0
</pre>
This last quotient, having remainder 0, indicates the polynomial factor  {{{x^2+2x+2}}}
and the zeros are found using general solution formula of a quadratic equation.


Zeros are  {{{(-2+- sqrt(2^2-4*2))/2}}}


{{{(-2+- sqrt(4-8))/2}}}


{{{(-2+- sqrt(-4))/2}}}


{{{(-2+- 2*sqrt(-1))/2}}}


{{{highlight(-1+- i)}}}


---------------------------------------------------------
To summarize all of these zeros-finding results,
REAL RATIONAL ZEROS:
-5 of multiplicity two;
-2
+2
COMPLEX ZEROS:
-1-i and -1+i
----------------------------------------------------------


To begin in forming the factorized form of your function, start with this, and then simplify the complex part if you want.
{{{highlight(f(x)=(x-(-5))^2(x-(-2))(x-2)(x-(-1-i))(x-(-1+i)))}}}