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 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 . You need to show that it is true for polynomials with degree .
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 is a root of a real polynomial, then also 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 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