Question 52578
HOW WOULD I USE SYNTHETIC DIVISION GIVEN THE POLYNOMIAL:
P(X)=X^4-4X^3-6X^2-4X-15 AND THE FACTOR THEOREM TO FIND WHETHER
X-(1+2i) IS A FACTOR
<pre><font size = 4><b>
Actually it's not as I will show you, but first it will be more 
instructive to you if I first demonstrate a case of a similar
polynomial for which x-(1+2i) is a factor:

Find whether x-(1+2i) is a factor of 

P(x) = x<sup>4</sup> - 6x<sup>3</sup> + 18x<sup>2</sup> - 30x + 25

We start out with the synthetic division algorithm. Start as 
usual by bringing down the 1:

1+2i| 1  -6     18     -30      25
    |<u>                             </u>   
      1                             

Multiply the 1 by 1+2i and put it diagonally above the 1 under 
the -6

1(1+2i) = 1+2i


1+2i| 1  -6     18     -30      25
    |<u>     1+2i                    </u>                     
      1                             

Add -6 and 1+2i, getting -5+2i, and write that on the bottom line


1+2i| 1  -6     18     -30      25
    |<u>     1+2i                    </u>                     
      1  -5+2i                    

Multiply the (-5+2i) by (1+2i)

(-5+2i)(1+2i) = -5-10i+2i+4iČ = -5-8i+4(-1) = -5-8i-4 = -9-8i

Write that diagonally above -5+2i

1+2i| 1  -6     18     -30      25
    |<u>     1+2i  -9-8i             </u>           
      1  -5+2i                     

Add 18 and -9-8i, getting 9-8i and write that on the bottom 
line


1+2i| 1  -6     18     -30      25
    |<u>     1+2i  -9-8i             </u>             
      1  -5+2i   9-8i              

Multiply 9-8i by 1+2i

(9-8i)(1+2i) = 9+18i-8i-16iČ = 9+10i-16(-1) = 9+10i+16 = 25+10i

Write that diagonally above 9-8i, then add it to -30 getting 
-5+10i

1+2i| 1  -6     18     -30      25
    |<u>     1+2i  -9-8i   25+10i    </u>     
      1  -5+2i   9-8i   -5+10i            

Multiply -5+10i by 1+2i

(-5+10i)(1+2i) = -5-10i+10i+20iČ = -5+20(-1) = -5-20 = -25

Write that diagonally above -5+25, add to 25 and get 0 remainder


1+2i| 1  -6     18     -30      25
    |<u>     1+2i  -9-8i   25+10i -25</u>
      1  -5+2i   9-8i   -5+10i   0

So we see that since we got a 0 remainder we can say that

x - (1+2i) IS a factor of x<sup>4</sup> - 6x<sup>3</sup> + 18x<sup>2</sup> - 30x + 25.

========================================================

Now if we do your problem the same way, we get

1+2i| 1  -4     -6     -4     -15
    |<u>     1+2i  -7-4i  -5-30i  51-48i</u>
      1  -3+2i -13-4i  -9-30i  36-48i

So as you see we do not get a 0 remainder, so NO, x-(1+2i)
is NOT a factor of P(x) = x<sup>4</sup> - 4x<sup>3</sup> - 6x<sup>2</sup> - 4x - 15.
as it was in the example I gave.  


Edwin</pre>