Can P(x) be factored into irreducible polynomials over Z_5?

[beetle2]

Homework Statement

Write $P(x) = x^3+2x+3$ as the product of Irreducible Polynomials over $Z_5$

Homework Equations

Polynomial division

The Attempt at a Solution

I start out by taking out a factor of $x+3$

That is

$x+3 \div x^3+2x+3$


I get $P(x) = x^2-3x+1$ which has zero remainder mod 5.


Is the product of irreducible polynomial $(x+3) (x^2-3x+1)$


or do I reduce $P(x) = x^2-3x+1$ by taking out a factor of x+1 ie


$x+1 \div x^2-3x+1$


I know the irreducible polynomials coefficients should add up to the original degree ,So I have one with degree 1 and the second with degree 2.

am I on the right track?

 

[Office_Shredder]

Well, a product of irreducible polynomials requires all your polynomials to be irreducible.

You're on the right track, the question now is: is x+3 irreducible, and is x²-3x+1 irreducible?

 

[beetle2]

I evaluated

$x+1 \div x^2-3x+1$

which is $P(X)= x-4$ zero remainder mod 5

So I have three irreducible Polynomials whose degrees add to three ie


$(x+3)(x+1)(x-4)$

Hows that look

 

[beetle2]

Is there a way to check that my answer is right?

 

[Dick]

Is there a way to check that my answer is right?

Multiply your product out and reduce the coefficients mod 5.

 

[beetle2]

I multiply it out and get

$x^3-13x-12$ which is $x^3-3x-2$mod 5 so I'm doing something wrong.

 

[Dick]

I multiply it out and get

$x^3-13x-12$ which is $x^3-3x-2$mod 5 so I'm doing something wrong.

Don't forget 2=(-3) and 3=(-2) mod 5.

 

[beetle2]

I think I need some more practice, It can get confusing just doing ordinary polynomial division without having modulo as well
$x^3+2x+3$

thanks for your help