Problem with polynomial division and 'i'

[Slicktacker]

I was learning polynomial division, and I can do most problems, except this one which is bothering me.

:

3x^2 + 2x + 7
---------------
(1+i)x - 2


How would I divide something like that? Nothing is working. Thanks.

[Muzza]

i is just another number, no different from 12 or pi or \sqrt{2}... You'd perform the long division like you usually do. The first term in the quotient is \frac{3}{1 + i}x, maybe that can get you started.

[HallsofIvy]

Another way to handle fractions involving compex numbers is to "realize" the denominator. (I just made up that word!) Multiply both numerator and denominator by the complex conjugate of the denominator: the complex conjugate of (1+i)x - 2 is (1-i)x- 2 (negative i instead of positive i). Multiplying the denominator (and numerator) by that gives you a fraction in which the numerator is a real number.

[jcsd]

You really want to mutiply through by the complex conjugate of 1+i tho'


edited to add Halls of Ivy beat me to it, again!

[Slicktacker]

I tried the problem and I got

3x - 1 ...... 3xi-i+6
------ remainder --------
1+ i ....... (1+i)x-2



Anybody know where I'm going wrong?

[jcsd]

What you want to do is mutply the top and bottom by he complex conjugate of 1 + i which is 1 - i to get:

\frac{(1-i)(3x^2 + 2x + 7)}{2x -4}

[Slicktacker]

I did that, and now I have a 1-i in the dividend, so how do I divide by that term using polynomial long division (should I multiply it by 3x^2+2x+7 ?)

[Slicktacker]

I get \frac{3}{2}x + 4 + \frac{23}{(2x-4)}, but where does the 1-i come in?

[Brad_Ad23]

Is the denominator (1+i)(x-2) or (1+i)x-2 ?

[Slicktacker]

The denominator is (1+i)x-2

[HallsofIvy]

\frac{3x^2+ 2x+ 7}{(1+i)x- 2}

The denominator is (1+i)x- 2 and its complex conjugate is (1-i)x- 2 (just replace i by -i).

Multiply both numerator and denominator by (1-i)x- 2

The numerator will become (1-i)x(3x2+ 2x+ 7)- 2(3x2+ 2x+ 7)
= (3-3i)x3+ (2-2i)x2+ (7-7i)x- 6x2- 4x- 14
= (3-3i)x3+ (-4-2i)x2+(3- 7i)x- 14

The denominator will become (1-i)x(1+i)x-2(1+i)x-2(1-i)x+ 4
= 2x2- 4x+ 4 (No i !!)

so the fraction is \frac{(3-3i)x^3+ (-4-2i)x^2+ (3-7i)x- 14}{2x^2- 4x- 4}

Now use long division to reduce that.

[JohnDubYa]

Of what use is polynomial long division (other than to find the leading-order behavior of a rational polynomial)?