over9000 questions thread

[Over9000]

KK, thanks that was actually the answer.
Another one, factorise the first expression into linear factors over C, given that the second expression is one of the linear factors

[TrueTears]

use long division.

hard to do on paint and i gotta go dinner now XD, if you got trouble long dividing it, i'll post it up later.

[Over9000]

Yeh, its okay, I get it now thanks

[kamil9876]

KK, thanks that was actually the answer.
Another one, factorise the first expression into linear factors over C, given that the second expression is one of the linear factors

I found this by accident:








I hate long division so i try to find creative ways of avoiding it.

[TrueTears]

KK, thanks that was actually the answer.
Another one, factorise the first expression into linear factors over C, given that the second expression is one of the linear factors

I found this by accident:








I hate long division so i try to find creative ways of avoiding it.

wow nice kamil, would expanding it out, generally yield your result?

[kamil9876]

Ummm....      not sure hahaha. But there is this technique that is actually on the course that is similair: It's where u look at the ratio of the coefficients to see if u can factorize in a neat way, I'll make up an example:

z^3 +2z^2 + 10z +20   

Notice the ratio 1:2 and 10:20 on the coefficients.

=z^2(z + 2) +10(z+2)
=(z^2+10)(z+2)

Although they're mostly more difficult but the same idea.
I actually tried to do this method for this question as i saw the potential of beautfiful ratios like 1:1:1  or -1:-1:-1. However it didn't quite worked out but a little tinkering in the end took me to this answer and i've never seen something like this.

It's hard to pick up, but it's something u should look for initially as it simplifies the problem a lot. (I can't even imagine doing long division... can u show us an example TT?)
It doesn't always come out this nicely, but it would be pretty crewl for it not too. Ussually for cubics.. it's either conjugate root theorem or this technique. Anythign else is crewl (equating coefficients(unless its easy) or long division  )

[kamil9876]

Here is an example from checkpoints. This one was on VCAA 2000 exam:

z^{3} -(2-i)z^{2} + z - 2 +i=0

a.) show that 2-i is a solution
b.)final all the solutions

Again. it;s the neat symmetry that u find that allows u to use this trick. It's the 1:-2:1 ratio for the first 3 and last 3 terms when u expand that allows u to factorise easily, as expected.

[Over9000]

Wow, thanks kamil 

[Over9000]

Sketch the following regions on an Argand diagram.

a)

I know how to do sketch but with modulus signs around it, how do i do it?

Thanks guys

[kamil9876]

if then .

THis is generally true for any case of a modulus. Say if |x|<2 then x can equal 1 or -1, 1.5 or -1.5, 1.99999999 or -1.99999999.

[Over9000]

Thanks alot kamil that really helped   :angel:  :angel:  :angel:  :angel:  :angel:

[Over9000]

Let {} {}. Let z = x+yi. (Hint: Use )

Find the relation between x and y such that:
a)
b)

c) In each of the two cases describe carefully the locus of z.

[kamil9876]

so for some number to be in . One obvious condition is that and so .

Another condition that must be satisfied is:

 

And when does that happen? You know that when you square a complex number the angle doubles. So we can half that set of angles:

 

However it could also be that:



Because of that thing you do with the argument when you take nth root of a complex number( adding on )

This should get you started.

Edit: also be aware that in my final inequality involving Arg(z), make sure u change the domain so that -pi<Arg(z)<pi

[Over9000]

How do i do this question?
This is the 1st part of the question
I am having troubles with the 2nd part and 3rd part

http://img16.imageshack.us/img16/2417/scan0001xec.jpg

2. The straight line that passes through Q and O can be described as the subset S of the complex plane, where {} and is a complex constant.
Find a possible w.

3). The chord of the circle K that passes through Q and O can be described as the intersection , where S is the subset defined above, and T is another subset of the complex plane.
Find a possible T.
(circle K just passes through the points P (-1,2i), Q(-1,-2i),  M(4,0) and D on the graph above)

[TrueTears]

1. sub and let w = x + yi

and then simplify, you should get . I think you can do it from there