Math 149. Capstone Mathematics for Teachers. Spring 2011.
Homework 8
Algebraic structure of the set of polynomials.
Let R[x] denote the set of all polynomials with real coefficients, with the usual addition and multiplication operations. Determine which of the following structures
this set has. If it has a certain structure, specify the identity and inverse elements for the required operations; do not write out proofs of properties such as
commutativity, asscociativity, and distributivity (these are lengthy but very straightforward). If it does not have a certain structure, identify at least one property
that does not hold.
Is R[x] a group under addition?
Is it a group under multiplication?
Is it a ring under addition and multiplication? If so, is it a commutative ring?
Is it a field under addition and multiplication?
Which number set (N, Z, Q, R, or C) has exactly the same algebraic structures as R[x]?
Finding a polynomial that has specified values.
Find a quadratic polynomial f(x) such that f(0)=2, f(1)=4, and f(2)=12.
(optional) Prove that for any distinct x1, x2, x3, and any y1, y2, y3, such that the points
(x1, y1), (x2, y2), and (x3, y3) do not lie on one line, there exists a quadratic polynomial f(x) such
that f(xi)=yi for i=1,2,3.