Problem with polynomial division and 'i'

In summary, the conversation is about polynomial long division and how to solve a specific problem using complex numbers. One method suggested is to multiply the numerator and denominator by the complex conjugate of the denominator, while another method is to use polynomial long division. The final answer is \frac{(3-3i)x^3+ (-4-2i)x^2+ (3-7i)x- 14}{2x^2- 4x- 4}.
  • #1
Slicktacker
9
0
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.
 
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  • #2
[tex]i[/tex] is just another number, no different from 12 or [tex]pi[/tex] or [tex]\sqrt{2}[/tex]... You'd perform the long division like you usually do. The first term in the quotient is [tex]\frac{3}{1 + i}x[/tex], maybe that can get you started.
 
  • #3
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.
 
  • #4
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!
 
  • #5
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?
 
  • #6
What you want to do is mutply the top and bottom by he complex conjugate of 1 + i which is 1 - i to get:

[tex]\frac{(1-i)(3x^2 + 2x + 7)}{2x -4}[/tex]
 
  • #7
I did that, and now I have a [tex]1-i[/tex] in the dividend, so how do I divide by that term using polynomial long division (should I multiply it by [tex]3x^2+2x+7[/tex] ?)
 
  • #8
I get [tex]\frac{3}{2}x + 4 + \frac{23}{(2x-4)}[/tex], but where does the [tex]1-i[/tex] come in?
 
  • #9
Is the denominator (1+i)(x-2) or (1+i)x-2 ?
 
  • #10
The denominator is [tex](1+i)x-2[/tex]
 
  • #11
[tex]\frac{3x^2+ 2x+ 7}{(1+i)x- 2}[/tex]

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 [tex]\frac{(3-3i)x^3+ (-4-2i)x^2+ (3-7i)x- 14}{2x^2- 4x- 4} [/tex]

Now use long division to reduce that.
 
  • #12
Of what use is polynomial long division (other than to find the leading-order behavior of a rational polynomial)?
 

FAQ: Problem with polynomial division and 'i'

What is polynomial division?

Polynomial division is a mathematical process used to divide one polynomial expression by another. It involves breaking down a polynomial expression into smaller, simpler parts to make it easier to solve.

What is the "i" in polynomial division?

The "i" in polynomial division refers to the imaginary unit, which is represented by the symbol √-1. It is used in complex numbers to represent the square root of a negative number.

Why is polynomial division with "i" important?

Polynomial division with "i" is important because it allows us to solve equations and problems that involve complex numbers. It is especially useful in fields such as engineering and physics, where complex numbers are commonly used.

How is polynomial division with "i" different from regular polynomial division?

Polynomial division with "i" is different from regular polynomial division because it involves using complex numbers and the rules of complex arithmetic. It also requires the use of the imaginary unit, which is not present in regular polynomial division.

What are some common challenges when working with polynomial division and "i"?

Some common challenges when working with polynomial division and "i" include understanding the rules of complex arithmetic, keeping track of the imaginary unit, and applying the correct steps in the division process. It is also important to be careful with signs and coefficients when simplifying complex expressions.

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