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!
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!
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?
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
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.
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