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spongegar
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I need to factor x5 - 1.I know (x-1) is a factor and have gotten:
(x-1)(x4+x3+x2+x +1)
I'm not sure where to go from here.
Thanks in Advance.
(x-1)(x4+x3+x2+x +1)
I'm not sure where to go from here.
Thanks in Advance.
spongegar said:Yes, I know that is a much more convenient approach and have used that to find the roots, but I need to factorize it for my assignment.
Thanks.
Yes, I know that is a much more convenient approach and have used that to find the roots, but I need to factorize it for my assignment.
If you are just wanting to factorize that quartic, you can input the coefficients here to see the solutions.spongegar said:I need to factor x5 - 1.
I know (x-1) is a factor and have gotten:
(x-1)(x4+x3+x2+x +1)
I'm not sure where to go from here.
Thanks in Advance.
Numerical approximations may be all that OP needs. He hasn't indicated.Mentallic said:There is a much better way of finding the roots than using that site which merely gives you numerical approximations.
NascentOxygen said:Numerical approximations may be all that OP needs. He hasn't indicated.
spongegar said:I need to factor x5 - 1
NascentOxygen said:Numerical approximations may be all that OP needs. He hasn't indicated.
Mentallic said:Yes he has
spongegar said:I need to factor x5 - 1
He may not have specifically said "I need to find the roots in exact form to factorize x5-1" but when you factor something, you don't do it with numerical approximations, you do it with exact roots - especially when the roots are relatively easy to find and it's a common question to be asked when studying complex numbers.NascentOxygen said:10 sig figs may be sufficient for OP's needs. He hasn't indicated.
I've seen numerical approximations to equations, but never have I seen them taken a step further and put into a factorized form.NascentOxygen said:The roots will give him the factors.
That's right, he didn't.Mentallic said:He may not have specifically said "I need to find the roots in exact form to factorize x5-1"
If the numerical approximations are all that's needed, then they may well suffice.but when you factor something, you don't do it with numerical approximations, you do it with exact roots - especially when the roots are relatively easy to find and it's a common question to be asked when studying complex numbers.
I think you might be satisfied with the 1.414 approximation, sometimes.For example, I wouldn't be satisfied with [tex]x^2-2\approx(x-1.414)(x+1.414)[/tex] when [tex](x-\sqrt{2})(x+\sqrt{2})[/tex] is readily available.
I made no mistake. You assumed he needed exact solutions, and that's fair enough. But I made no such assumption. OP's wording had a nuance that hinted his need may be part of an assignment, not the totality of it. Were this to be a case where he'd end up using the numerical values anyway, he may not even need to spend time evaluating the exact factors. I pointed out his options were open; OP can make the judgement on what suits his need.Mentallic said:If you can show me one example of where numerical approximations to roots of polynomials have been put in factored form, then you have a case. Until then, it just looks to me as though you're desperately grabbing at straws to try back up what you initially said.
I'm not here to point the finger and laugh at your honest mistake, I'm only trying to set the facts straight that if an assignment asks for a polynomial to be factored, then it must be in exact form. If it asked for the solutions however, then they may be numerical approximations.
A root of unity is a complex number that, when raised to a certain power, results in a value of 1. For example, the numbers 1, -1, i, and -i are all roots of unity because they have a power of 1 that equals 1.
To factor a root of unity, you must first identify the value of the root and its corresponding power. Then, you can use the formula (x - a)(x - a^2)(x - a^3)...(x - a^n) = 0, where a is the value of the root and n is the power. This will give you the factors of the root of unity.
Factoring a root of unity is important in understanding complex numbers and their properties. It also has applications in fields such as number theory, algebraic geometry, and physics.
Yes, all roots of unity can be factored using the formula mentioned in question 2. However, some roots may have repeated factors, such as (x - 1)^2 or (x + i)^3.
Roots of unity are used in various mathematical concepts and equations, such as in polynomial equations, trigonometric functions, and Fourier series. They also have applications in fields like number theory, abstract algebra, and complex analysis.