Factorize polynomial

  • Thread starter cdummie
  • Start date
147
5
I should factorize following polynomial:

P(x)=x^2n + 2cos(naπ)x^n + 1 in ℝ if i know that a is irrational number.

Things that confuse me here are following:

1. When factorizing polynomials, i have known exponents (unlike here, where i have 2n and n) so i don't know what to do with them?

2. Why does it makes a difference if a is irrational number?
 
10,823
4,353
Have you considered a substitution of variable x to make the equation a simple quadratic?
 
147
5
Have you considered a substitution of variable x to make the equation a simple quadratic?
You mean like: y=xn, but what can i do next, that cosine there is confusing, and fact that a is irrational makes it even more complicated.
 

SteamKing

Staff Emeritus
Science Advisor
Homework Helper
12,794
1,665
You mean like: y=xn, but what can i do next, that cosine there is confusing, and fact that a is irrational makes it even more complicated.
The value of the cosine in your polynomial doesn't depend on x, so it may be treated like a constant. Call it A and re-write your original polynomial:

A = cos(naπ)

P(x) = x2n + 2Axn + 1

Now, make the substitution y = xn
 
147
5
The value of the cosine in your polynomial doesn't depend on x, so it may be treated like a constant. Call it A and re-write your original polynomial:

A = cos(naπ)

P(x) = x2n + 2Axn + 1

Now, make the substitution y = xn

Ok, so now i have the following:

y2 + 2Ay + 1

y1,2=[-2A+-sqrt(4A2 - 4)]/2 = -A +- sqrt(A2 - 1)

since A=cos(naπ) then A2=cos2(naπ) and A2 - 1 = cos2(naπ) - 1 =-sin2(naπ)

but -sin2(naπ) could be anywhere between -1 and 1 so i don't know is it positive or negative, because if it is negative then i can't factorize it in ℝ.
 

SteamKing

Staff Emeritus
Science Advisor
Homework Helper
12,794
1,665
Ok, so now i have the following:

y2 + 2Ay + 1

y1,2=[-2A+-sqrt(4A2 - 4)]/2 = -A +- sqrt(A2 - 1)

since A=cos(naπ) then A2=cos2(naπ) and A2 - 1 = cos2(naπ) - 1 =-sin2(naπ)

but -sin2(naπ) could be anywhere between -1 and 1 so i don't know is it positive or negative, because if it is negative then i can't factorize it in ℝ.
Obviously, in order to factor this polynomial in ℝ, certain conditions must be met; IOW, A2 - 1 ≥ 0, or A2 ≥ 1 or cos2(naπ) ≥ 1.

Since -1 ≤ cos (θ) ≤ 1, then the polynomial can be factored in ℝ only if naπ = 2kπ, where k = 0, 1, 2, ...

It's not clear why you would want to factor such a beast anyway. If you want to find the roots to P(x), it would seem that a numerical approach would work just as well.
 
147
5
Obviously, in order to factor this polynomial in ℝ, certain conditions must be met; IOW, A2 - 1 ≥ 0, or A2 ≥ 1 or cos2(naπ) ≥ 1.

Since -1 ≤ cos (θ) ≤ 1, then the polynomial can be factored in ℝ only if naπ = 2kπ, where k = 0, 1, 2, ...

It's not clear why you would want to factor such a beast anyway. If you want to find the roots to P(x), it would seem that a numerical approach would work just as well.
Well, it's just an exam question, anyway, what's with a, i mean why is it pointed out that a has to be irrational number, the way we solved this, it wouldn't be any different even if a wasn't irrational number?
 

SteamKing

Staff Emeritus
Science Advisor
Homework Helper
12,794
1,665
Well, it's just an exam question, anyway, what's with a, i mean why is it pointed out that a has to be irrational number, the way we solved this, it wouldn't be any different even if a wasn't irrational number?
It's not clear what n is here. Is n any integer, or is it something else?

Since a is irrational, this suggests that the condition naπ = 2kπ can be satisfied only if na = 0, and this will depend on what values n can take.
 
147
5
It's not clear what n is here. Is n any integer, or is it something else?

Since a is irrational, this suggests that the condition naπ = 2kπ can be satisfied only if na = 0, and this will depend on what values n can take.
I don't know about n, it's not defined what n is, but it might be integer (probably it is), anyway, since it is not defined i can't be sure what it is.
 

Related Threads for: Factorize polynomial

  • Posted
Replies
1
Views
2K
  • Posted
Replies
1
Views
450
  • Posted
Replies
5
Views
2K
  • Posted
Replies
7
Views
2K
  • Posted
Replies
6
Views
1K
  • Posted
Replies
13
Views
6K
  • Posted
Replies
15
Views
2K
  • Posted
Replies
11
Views
2K

Physics Forums Values

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving

Hot Threads

Top