1. ## Irreducible Polynomial

What makes a polynomial irreducible?

2. ## Re: Irreducible Polynomial

Have you looked at the definition?

3. ## Re: Irreducible Polynomial

A polynomial is "irreducible" if it can't be reduced!

Now, what are you really asking?

4. ## Re: Irreducible Polynomial

Originally Posted by HallsofIvy
A polynomial is "irreducible" if it can't be reduced!

Now, what are you really asking?
For example, why can x^4 + 64 be reduced any further?

5. ## Re: Irreducible Polynomial

Originally Posted by chiro

Have you looked at the definition?
No. Reduce x^3 + 9 to the lowest terms. It cannot be reduced. Why?

6. ## Re: Irreducible Polynomial

Basically, a non-constant polynomial is irreducible if it cannot be factorised into non-constant polynomials.

(An example of a non-constant polynomial is $P(x)=x^2 + 3x+2$. An example of a constant polynomial (sometimes called trivial) is $P(x) = 3$.)

7. ## Re: Irreducible Polynomial

I'm not seeing anyone touch an the real (pun intended) answer.

Polynomials are never irreducible. Over the complex numbers.

They are only irreducible over the reals (or rationals, integers etc.).

A polynomial that is irreducible over the reals is one that has strictly complex roots. (note this will be an even degree polynomial. why? )

what are the roots of $x^4 + 64$ ?

w/o going into detail these are $x=\pm 2 \pm 2 i$

i.e. all the roots are complex

8. ## Re: Irreducible Polynomial

Originally Posted by romsek
I'm not seeing anyone touch an the real (pun intended) answer.

Polynomials are never irreducible. Over the complex numbers.

They are only irreducible over the reals (or rationals, integers etc.).

A polynomial that is irreducible over the reals is one that has strictly complex roots. (note this will be an even degree polynomial. why? )

what are the roots of $x^4 + 64$ ?

w/o going into detail these are $x=\pm 2 \pm 2 i$

i.e. all the roots are complex
Sure you're correct Romsek. Didn't want to get too complex (pun intended) for Mathdad just yet.

9. ## Re: Irreducible Polynomial

Originally Posted by Debsta
Basically, a non-constant polynomial is irreducible if it cannot be factorised into non-constant polynomials.

(An example of a non-constant polynomial is $P(x)=x^2 + 3x+2$. An example of a constant polynomial (sometimes called trivial) is $P(x) = 3$.)
I had no idea that P(x) = 3 is a polynomial. Why do they call it trivial?

10. ## Re: Irreducible Polynomial

Originally Posted by Debsta
Sure you're correct Romsek. Didn't want to get too complex (pun intended) for Mathdad just yet.
I appreciate your patience with me. I haven't sat through a formal math class since 1994 but everything is slowly but surely coming back to me.

11. ## Re: Irreducible Polynomial

I had no idea that P(x) = 3 is a polynomial. Why do they call it trivial?
Well it meets the definition of a polynomial, but contains no variable I suppose.

12. ## Re: Irreducible Polynomial

I appreciate your patience with me. I haven't sat through a formal math class since 1994 but everything is slowly but surely coming back to me.
No worries. Love helping people like you who are making the effort!

13. ## Re: Irreducible Polynomial

For example, why > > can < < x^4 + 64 be reduced any further?
Originally Posted by romsek

A polynomial that is irreducible over the reals is one that has strictly complex roots.
(note this will be an even degree polynomial. why? ) **

what are the roots of $x^4 + 64$ ?

w/o going into detail these are $x=\pm 2 \pm 2 i$

i.e. all the roots are complex
But $x^4 + 64$ factors into $(x^2 - 4x + 8)(x^2 + 4x + 8).$ That means it is
reducible over the reals. In particular, it is reducible over the integers.

** From above in the lower quote box. This polynomial has strictly complex
roots, but it is not irreducible.

14. ## Re: Irreducible Polynomial

Originally Posted by greg1313
But $x^4 + 64$ factors into $(x^2 - 4x + 8)(x^2 + 4x + 8).$ That means it is
reducible over the reals. In particular, it is reducible over the integers.

** From above in the lower quote box. This polynomial has strictly complex
roots, but it is not irreducible.
you're correct. Please ignore my post regarding this.

15. ## Re: Irreducible Polynomial

Also, any polynomial (of a single variable) of order 1 is irreducible.

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