Complex Number Leaving Cert Question

dioltas

From the leaving cert honours maths 2003 paper 1, question 3.

1, w, w^2 are the three roots of the equation z^3 - 1 = 0.

(i) Prove 1 + w + w^2 = 0.

Does anyone know how to solve this? I assume w and z are complex numbers? Been a while since I've done something like this, trying to help brother with leaving. Thanks.

LeixlipRed

Have a look at this article here and see if you can grasp it. Come back to us if it's still not clear. EDIT: That article has no example. Look at this one here.

RoundTower

I guarantee this exact question is in the textbook.

LeixlipRed

Yeh that's pretty certain

rjt

Another way of looking at this, can you factor z^3 - 1? Just use the a^n - b^n formula. You get two factors, one of which has to be zero...

dioltas

Thanks, got it now. Was kinda going down the polar form route alright, but was really overcomplicating it a bit. I agree it's probably in the book, brother just asked me how to do it and was a bit put out that I couldn't figure it out! He's doing english tomorrow anyway, so he's not concerned with maths anymore!

Thanks for the help.

LeixlipRed

rjt wrote: »

Another way of looking at this, can you factor z^3 - 1? Just use the a^n - b^n formula. You get two factors, one of which has to be zero...

Hmmm, do you mean equal to zero?

rjt

LeixlipRed wrote: »

Hmmm, do you mean equal to zero?

Yeah. I guess that saying "0 is a factor" is a bit odd. Also, by "the a^n - b^n formula" I meant:

a^n - b^n = ( a - b ) ( a^(n-1) + a^(n-2) b + ... + a b^(n-2) + b^(n-1) )

I'd forgotten that they probably don't cover this for the LC, so my post probably wasn't terribly useful.

RoundTower

rjt wrote: »

Yeah. I guess that saying "0 is a factor" is a bit odd. Also, by "the a^n - b^n formula" I meant:

a^n - b^n = ( a - b ) ( a^(n-1) + a^(n-2) b + ... + a b^(n-2) + b^(n-1) )

I'd forgotten that they probably don't cover this for the LC, so my post probably wasn't terribly useful.

this is on the leaving cert (at least for n=3)

nice one-liner solution to the problem

MathsManiac

OP, if you're helping out the brother with more material, it may be useful to know that marking schemes, which include full solutions to all questions, are available at the State xams Commission's website: www.examinations.ie, under the heading "examination material archive".

The marking scheme for 2003 gives four different solutions to that particular question:
http://www.examinations.ie/archive/markingschemes/2003/LC003ALP1EV.PDF

dioltas

MathsManiac wrote: »

OP, if you're helping out the brother with more material, it may be useful to know that marking schemes, which include full solutions to all questions, are available at the State xams Commission's website: www.examinations.ie, under the heading "examination material archive".

The marking scheme for 2003 gives four different solutions to that particular question:
http://www.examinations.ie/archive/markingschemes/2003/LC003ALP1EV.PDF

Thanks for that. Should come in handy. He's Maths paper 1 this morning. Was helping him out a bit last night, think I might have just confused him more than anything though!