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RJLiberator
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Homework Statement
1. Let g(x) = x^4+46.
a) Factor g(x) completely in ℚ[x].
b) Factor g(x) completely in ℝ[x].
c) Factor g(x) completely in ℂ[x].
2. Completely factor the given polynomial in ℤ_5.
[4]_5 x^3 + [2]_5 x^2 + x + [3]_5
Homework Equations
ℚ = {m/n / m and n belong to Z, m is not divided by m}
ℝ = {real numbers which include all rational and irrational numbers}
ℂ = Complex
A polynomial is completely factored over F if it is written as a product of monic irreducible polynomials in F[x] and an element of F.
The Attempt at a Solution
Initial thoughts: This looks easy! However, it's a bit tougher than it looked.
We have x^4+46.
To start, let's just factor it as I normally would without worry about the fields.
With a little help from wolfram alpha, and a personal check to make sure we see that for the factorization over ℂ (part c) we get:
(x+4throot(-46)) (x-4throot(-46)) (x-i*4throot(-46)) (x+i*4throot(-46))
This should be the correct answer to c as it is in completely factored form.
Now, we try to get the factorization over the reals, ℝ. part b:
Since we can't use i, we try to multiply the factors a bit to get rid of i.
We get : (x^2-4throot(-46)) (x^2+4throot(-46))
Question: Is this ok over field ℝ? I have a negative under the square root, but I'm not using imaginary unit i.
Answer: Likely not as this is an expression of a+ib.
So, this doesn't work.At this point, I'm kind of lost. I feel like, over the field ℚ, the polynomial g(x) is completely factored.
But I see to have come to the conclusion that over the field ℝ it is also completely factored.
My final answers:
1.a It is already completely factored
1.b It is completely factored
1.c I factored it above.
As for part 2, the difficulty here is dealing with the mod 5.
If we look at it like 4x^3+2x^2+x+3
I would think it would be in the form:
(x+)(x+)(x+)
But I'm not really sure on how to start this one. I understand what the mod 5 does, but I'm not used to trying to factor x^3+x^2+x+a.