Abstract Algebra: Theory and Applications · Problem 17.5.3

Integral Domains — Thomas Judson

Mathematics — Abstract Algebra · polynomial-division-algorithm

Problem

Use the division algorithm to find

q(x)

and

r(x)

such that

a(x) = q(x)b(x) + r(x)

with

\deg r(x) < \deg b(x)

for each of the following pairs of polynomials. ((a))

a(x) = 5x^3 + 6x^2 - 3x + 4

and

b(x) = x - 2

\mathbb{Z}_7[x]

((b))

a(x) = 6x^4 - 2x^3 + x^2 - 3x + 1

and

b(x) = x^2 + x - 2

\mathbb{Z}_7[x]

((c))

a(x) = 4x^5 - x^3 + x^2 + 4

and

b(x) = x^3 - 2

\mathbb{Z}_5[x]

((d))

a(x) = x^5 + x^3 - x^2 - x

and

b(x) = x^3 + x

\mathbb{Z}_2[x]

Hint. (a)

5x^3 + 6x^2 - 3x + 4 = (5x^2 + 2x + 1)(x-2) + 6

; (c)

4x^5 - x^3 + x^2 + 4 = (4x^2+4)(x^3+3) + 4x^2 + 2