Polynomials

Question 1

What is a polynomial

Answer 1

Polynomial is a0+ A1x+…+ AnX^n

Question 2

What is degree of 0 polynomial

Answer 2

Degree of 0 polynomial is - inf

Question 3

When is a polynomial Monic

Answer 3

Polynomial is monic when leading coefficient = 1 Xn

Question 4

What is a ring

Answer 4

A ring is where there is commutative addition and scalar multiplication and have 1 and 0

Question 5

What are degrees of polynomials in a ring

Answer 5

Degrees of polynomials in a ring are:
Det pq = det p + deg q
Deg(p+q) <= max(deg p, deg q)

Question 6

What is evaluation of polynomial p = a0+ …+ AnX^n

Answer 6

Evaluation of polynomial p= a0 +…cAnX^n is
P(t) = A0 + A12 +…+Ant^n

Question 7

What does evaluation preserve

Answer 7

Evaluation preserves addition and multiplication, (p+q)(t) = p(t) + q(t) (pq)(t) = p(t)q(t) so p to p(t) is a linear map
Defined function

Question 8

If p,q are in F([x], then how can p be written

Answer 8

If p,q F[x] then p can be written as
P= s*q + r with unique r,s in F[x] and deg r < deg q

Question 9

What is fundamental theorem of algebra

Answer 9

Fundamental theorem of algebra is every polynomial has a root in C, polynomial of deg n has at most n roots

Question 10

How can polynomials be written in C

Answer 10

In C, polynomials can be written as p = a*PI(x - Li)^ni
A is leading coefficient, Ali are roots, n is multiplication