Factorising this damned equation

[StephenP91]

Well, it's only Pure Core 1 Mathematics. I am trying to factorise:

2x^3 - 6x^2 + 2 = 0

Now, you can't use a calculator. I've tried finding a factor using the remainder theorem, but I just can't find a simple one (|x<5|). I am sure they don't expect us to use a complicated number, like a non-integar. So I just need someone to help me with factorising this.

Thank you,
Stephen.

 

[Hurkyl]

Remainder theorem?

There is the rational root theorem which gives you a short list of things to try -- and if your polynomial has any rational roots, they must appear on this list.


That said, it's "easy" to see that this polynomial doesn't have any linear factors. Can you give an argument that it doesn't have any quadratic factors?

However, I guarantee that this polynomial has a nontrivial factor... so if it's not linear, and it's not quadratic, what must it be?

 

[Hurkyl]

P.S. by "factorize", I assume you mean to factor over the integers -- i.e. you want each factor to be a polynomial with integer coefficients.

 

[StephenP91]

By factorise I mean, place intro brackets so that I may find the information I am looking for. Namely where the graph crosses the X axis so that I can plot the graph.

I was thinking though. Could I just factorise 2x^3 - 6x^2 into 2x^2(x-3) and then get the points I need, then from that subtract 2 to each of the y co-ords to get the co-ords of each of the roots?