View Full Version : Maths help please! (finding roots of a degree 5 polynomial) +REP
Matthew
24-10-2013, 11:19 AM
Basically, I've got to factorise 16x5 - 20x3 + 5x - 1 = 0
So far I just used the factor theorem to find (x-1) to be a root, and then divided the above polynomial by (x-1).
This left me with 16x4 + 16x3 - 4x2 - 4x + 1
where do I go from here? :(
this is FP3 by the way!
@Kardan (http://www.habboxforum.com/member.php?u=3428); as you normally seem to be able to answer my questions haha
I'm probably missing something super obvious but I've just stared at it for 10mins :P
Kardan
24-10-2013, 12:13 PM
First step is correct.
We now have 16x^4 + 16x^3 - 4x^2 - 4x + 1
We can represent this equation as a quadratic equation squared, so we will have to find (ax^2 + bx + c)^2 = 16x^4 + 16x^3 - 4x^2 - 4x + 1
Multiplying out the bracket gives us: (ax^2 + bx + c)(ax^2 + bx + c) = (a^2)x^4 + abx^3 + acx^2 + abx^3 + (b^2)x^2 + bcx + acx^2 + bcx + c^2
Collate coefficients together:
a^2(x^4) + 2ab(x^3) + (2ac + b^2)x^2 + (2bc)x + c^2
Equate this to our degree 4 polynomial:
a^2(x^4) + 2ab(x^3) + (2ac + b^2)x^2 + (2bc)x + c^2 = 16x^4 + 16x^3 - 4x^2 - 4x + 1
Equate coefficients:
a^2(x^4) = 16(x^4), so a = 4.
2ab(x^3) = 16(x^3), so 2ab = 16, sub in a = 4 to give 8b = 16, so b = 2.
2bc(x) = -4x, so 2bc = -4, sub in b = 2 to give 4c = -4, so c = -1.
Check that a, b and c are correct with x^2 coefficient:
(2ac + b^2)x^2 = -4(x^2)
So 2ac + b^2 = -4, sub in a = 4, b = 2 and c = -1 to give 2*4*-1 + 4 = -4 as needed.
So our 4 degree polynomial can be expressed as (4x^2 + 2x - 1)^2
Can you solve it from here? :)
Matthew
24-10-2013, 12:24 PM
-snip-
Can you solve it from here? :)
I've never come across that method before so I wouldn't know how to, no :(
Maybe I should have posted the question itself- we're currently using De Moivre's Theorem.
http://i.imgur.com/IA2b7q1.jpg
I've managed to part (i) as it boiled down to a simple hidden quadratic but the '1' being there on part (ii) is what's confusing me!
Kardan
24-10-2013, 12:38 PM
With (4x^2 + 2x - 1)^2 = 0, take the square root of both sides, giving you:
4x^2 + 2x - 1 = 0 - Simply solve the quadratic for the roots.
That's what I would do at least, if there's a certain method to it that also involves polynomials, I can't see it at all.
Matthew
24-10-2013, 12:43 PM
With (4x^2 + 2x - 1)^2 = 0, take the square root of both sides, giving you:
4x^2 + 2x - 1 = 0 - Simply solve the quadratic for the roots.
That's what I would do at least, if there's a certain method to it that also involves polynomials, I can't see it at all.
Yeah I saw that a minute or so after posting!
Thanks for your help, I've certainly never had to factorize a polynomial like this before :O
I mean the second part of the homework was to do the same sorta thing but with sin6x = 0 (finding it in terms of cosx and sinx yourself), which again is easy as its just a hidden quadratic and DMT is simple anyway. Ah well, thanks again!
Shockwave.2CC
25-10-2013, 01:20 PM
So glad I didn't have to do this
This is bloody confusing :s
Matthew
25-10-2013, 03:21 PM
So glad I didn't have to do this
This is bloody confusing :s
well i'd imagine it would be if you hadn't learnt/seen it before :P
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