Can a monic polynomial be negative? (2 Viewers)

[sungkwo]

Well?

 

[sungkwo]

monic = degree of 1

so yes.

No, I mean as in the leading term having a coefficient of one.

 

[wendus]

yeah monic means leading coefficient is 1
- 1 =/= 1

 

[xXmuffin0manXx]

yes

edit:actually my bad..im not sure

 

[kaz1]

y=-x² is monic and negative.

 

[cHoke-meh]

Wait, i'm getting confused now. When you say monic, does the coefficient of x have to be positive 1? or just +_ 1?

 

[suzlee]

Wait, i'm getting confused now. When you say monic, does the coefficient of x have to be positive 1? or just +_ 1?

According to kaz1, and I think he's right, as long as the leading term has a coefficient of either positive or negative 1, the poly is a monic.

 

[supercalamari]

*Scratches head and wonders which of the hundreds of year 9 and 10 maths lessons I slept through contained info about monic polynomials*

Why don't you google it.

 

[ratcher0071]

Well technically:
Can a monic polynomial be negative? Yes it can.
A monic polynomial has the leading coefficient of 1. If you put a negative sign in front you have made a monic polynomial, negative.

 

[ubermale]

Well technically:
Can a monic polynomial be negative? Yes it can.
A monic polynomial has the leading coefficient of 1. If you put a negative sign in front you have made a monic polynomial, negative.

True, you can make a monic polynomial negative, but once the leading coefficient is not 1 then it is no longer monic. Therefore, if a polynomial had a leading coefficient of -1, it would be considered non-monic.

 

[Iruka]

A monic polynomial has a leading coefficient of 1, not -1.

You may wonder why this is the case, but it is defined that way to make some of the theorems simpler.

For example, when you define the GCD (greatest common divisor) of 2 polynomials, you want to get a unique answer, so the GCD of two polys (as opposed to just a GCD of two polys) is defined as the monic polynomial of greatest degree that divides both the polynomials evenly.

http://en.wikipedia.org/wiki/Greatest_common_divisor_of_two_polynomials

If polynomials with a negative leading coefficient were allowed to be monic, then you still wouldn't have a unique GCD.