Expert: Josh - 4/24/2006

Question
I need to find the answer to three questions- these ones I am struggling with
They are all T and F w/ an explanation needed.
1)  product of monic polynomials monic?
2)  product of polynomials of degrees m and n has
degree m & n.
3) The sum of polynomials of degree m and n has degree max (m,n)

Answer
1. A monic polynomial has a leading coefficient of 1.
Some examples include x^2+2x+1; x^3-2x+4; x^6-1.
The question asks when you multiply two monic polynomials together, do you always get a leading coefficient of 1?
Answer is yes. To see this, consider question 2.

2. In general, let
p(x)=x^n+a[n-1]*x^(n-1)+a[n-2]*x^(n-2)+...+a[0];
q(x)=x^m+b[m-1]*x^(m-1)+b[m-2]*x^(m-2)+...+b[0]
be two monic polynomials of degree n and m, respectively. The a[i], b[i] terms are the coefficients corresponding to x^i in p(x) and q(x).
The resultant polynomial p(x)*q(x) must be of the form x^(n+m)+...some lower order terms.
So, p(x)*q(x)
(i) must be a monic polynomial;
(ii)will have degree m+n (not m & n).

3. True. I won't give a proof, but just look at an example.
Consider p(x)=x^4+3x^3-1 and q(x)= x^3+x^2+6.
p(x)+q(x)= x^4+4x^3+x^2+5.
Here, p(x) has degree m=4, q(x) has degree n=3.
The resultant polynomial cannot exceed order max(m,n)=max(4,3)=4.