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
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) = x4 - 6x3 + 18x2 - 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
|
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
| 1+2i
1
Add -6 and 1+2i, getting -5+2i, and write that on the bottom line
1+2i| 1 -6 18 -30 25
| 1+2i
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
| 1+2i -9-8i
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
| 1+2i -9-8i
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
| 1+2i -9-8i 25+10i
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
| 1+2i -9-8i 25+10i -25
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 x4 - 6x3 + 18x2 - 30x + 25.
========================================================
Now if we do your problem the same way, we get
1+2i| 1 -4 -6 -4 -15
| 1+2i -7-4i -5-30i 51-48i
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) = x4 - 4x3 - 6x2 - 4x - 15.
as it was in the example I gave.
Edwin