- #1

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.