Can someone please help me with this question:

Prove by induction that every polynomial in R[x] can be written as a product of irreducible polynomials in R[x] (Note: R is the real numbers)

Thank you!

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- Jan 6th 2010, 04:11 PMpseudonymPolynomials written as a product of irreducible polynomials
Can someone please help me with this question:

Prove by induction that every polynomial in R[x] can be written as a product of irreducible polynomials in R[x] (Note: R is the real numbers)

Thank you! - Jan 6th 2010, 05:43 PMShanks
Hint: prove by induction on the degree of polynomial in R[x].

- Jan 7th 2010, 04:07 AMHallsofIvy
- Jan 7th 2010, 07:46 PMRoam
Here's a hint:

**Base case**: for $\displaystyle n=1$ every polynomial of degree 1 is irreducible so it can be written as itself.

**Inductive step**: We can assume the statement is true for polynomials with degree $\displaystyle 1,...,n$. You need to show that it is true for polynomials with degree $\displaystyle n+1$.

Can you continue from here? - Jan 7th 2010, 08:58 PMtonio

Two hints more (well read and better thought, the following hints are about 90% of the answer to your question):

1) Any real polynomial of odd degree has a real root (m.v.t. for continuous functions) , and:

2) If $\displaystyle z\in\mathbb{C}$ is a root of a real polynomial, then also $\displaystyle \overline{z}$ is , or in other words: complex non-real roots of a real polynomial come in conjugate pairs.

Tonio - Jan 8th 2010, 12:46 AMSwlabr
- Jan 8th 2010, 01:52 AMRoam
To prove the inductive step you have to note that all polynomials of degree $\displaystyle n+1$ are either irreducible (if so they can be written as a themselves), or written as a product of two polynomials with smaller degree. Then you can use the assumption and write down each of them as a product of irreducible polynomials.

- Jan 8th 2010, 03:13 AMtonio

As many other times in the past I misread, or better: didn't read completely. The induction proof works for polynomials over any integral domain, and since I read "real" I assumed (because, and I checked, I did NOT read it in the OP) that he meant : any real polynomial can be written as the product of irreducible polynomials....of degree 1 and/or 2 .

My hints, thus, as huge overkill.

Tonio