Polynomials Question (1 Viewer)

[KFunk]

This question (no.10 from further questions in the arnold & arnold) has been gving me grief so I figured I'd post it here to let you guys have a crack at it. I'm sure I must have overlooked something. Here it is:

The equation x³ + 3px² + 3qx + r = 0, where p²≠q, has a double root.

Show that (pq - r)² = 4(p² - q)(q² - pr).

 

[KFunk]

Maybe this is just me being hopeful but... Bump.

 

[Slidey]

I'll be honest with you - I tried a large variety of algabraic manipulations, substitutions, and anything else you care to name, last night. In a few cases I almost got somewhere.

I bet it involves a very obscure algabraic manipulation such that if you didn't get it the first time, you wouldn't be likely TO get it.

I doubt they'd ask anything that hard in an exam.

 

[Ogden_Nash]

Yeah, I've seen the solution to this question and it involves coming up with a lot of identities and manipulating them to get the answer.

I doubt they'd ask anything that hard in an exam.

Agreed.

 

[Estel]

I just had a look, got the paper out, and managed it first try ... slice of luck
It's actually under a page, and I haven't studied polynomials, so I believe the art is to break down the question into parts you can manage:
p = -(2a+b)/3
q = a(a+2b)/3
r = -a^2b
and the rest really just follows...

 

[KFunk]

Thanks for trying ,
I had a go at a whole bunch of things too and the thing which seemed most on track was taking the first derivative ('cause there's a double root) which is a quadratic and using the quadratic equation. Under the square root you end up with (p² - q) which incurs that statement which was at the begining of p²≠q. Plus-minuses always scare me and I didn't get very far with it .

 

[Estel]

Don't get fancy:
LHS = 4a/81 (a-b)^4
RHS = 4a/81 (a-b)^4 = LHS

I am, however, on the watch for a more elegant solution.

 

[KFunk]

Don't get fancy:
LHS = 4a/81 (a-b)^4
RHS = 4a/81 (a-b)^4 = LHS

I am, however, on the watch for a more elegant solution.

Bravo, very neatly done.

 

[Estel]

I should add, btw, that I set a, a and b to be the roots.
Otherwise my statement is quite useless.
In fact, it's completely useless

 

[KFunk]

In fact, it's completely useless

Worked for me . well.. provided you make it 4a² on 81.

 

[Estel]

Kfunk: yup, that's what I got but for some reason I didn't type the squared... hmmmmmm

I am quite bored.

 

[Slidey]

Root-coefficient.. you're kidding me. I saw double root and just ignored everything that didn't involve the first derivative. Bah humbug.

EDIT: First derivative, not second.

 

[Emma-Jayde]

I doubt they'd ask anything that hard in an exam.

I can't say that we should count on that. This is extension 2, they're bound to ask us complicated questions!

 

[FinalFantasy]

hey emma jayde r u the user Jaydels?

 

[Slidey]

Of course. I thought this question was far harder than it actually is.

 

[Emma-Jayde]

hey emma jayde r u the user Jaydels?

Nope, I'm not!

Hey, this quote from my maths teacher might help solving some 4 unit questions:

"Maths hint number 6749: If it looks like something you haven't seen before, you really have."

 

[FinalFantasy]

lol
(blah blah blah to make up 10 letters)

 

[KFunk]

haha
(spam spam spam spam, spam spam spam spam)