Page 227 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 2. The degree of the zero polynomial to be ................. thus degree 0 = .................
(a) – , –  (b) , 

(c) , –1 (d) –1, 
3. 3x + 4x + 5 is a polynomial of degree ................., whose coefficients belong to the ring of
2
integers Z its leading coefficients is .................
(a) 4, 5 (b) 2, 3
(c) 2, 4 (d) 2, 5
4. x + 2x + 6x + 8 is a polynomial of degree ................. with coefficient 2.
4
2
(a) 4 (b) 5
(c) 6 (d) 8
5. Let R be a ring and r  R, R ................. 0. Then r is polynomial degree of 0 with leading
coefficient r.

(a) = (b) 
(c)  (d) 

22.3 Summary

The definition and examples of polynomials over a ring.

The ring structure df R[x], where R is a ring.

R is a commutative ring with identity iff R[x] is a commutative ring with identity.

R is an integral domain iff R[x] is an integral domain.

22.4 Keywords

Polynomial: A polynomial over a ring R in the indeterminate x is an expression of the form
a x + a x + a x + ... + a x ,
2
0
1
n
0
n
1
2
where n is a non-negative integer and a , a,, ..., a  R.
n
0
Coefficient of Polynomial: Let a + a, x + ... + a, x be a polynomial over a ring R. Each of a ,a , .
n
0
n
l
. ., a, is a coefficient of this polynomial. If a,  0, we call a, the leading coefficient of this
polynomial.
22.5 Review Questions
1. Identify the polynomials from the following expressions. Which of these are elements of
Z[x]?
2 1
(a) x + x + x + x + x + 1 (b)   x x 2
2

5
4
6
x 2 x
1 1 1
2
2
(c) 3x  2x  5 (d) 1  x  x  x 3
2 3 4
(e) x + 2x + 3x 5/2 (f) –5
3/2
1/2
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