What is the Remainder of Polynomial Division in Z5[x] by x+3?

[sarah77]

Homework Statement

Find the remainder of x^4 + 3x +2 after division by x+3 in Z5[x]

Homework Equations

my quotient after dividing was: x^3 + 2X^2 + 4x +1

The Attempt at a Solution

I found the remainder to be 4. If anyone has time, I believe I made a mistake somewhere and would like someone to attempt this division so I can check my work. If you have time please check it because I am not confident in dividing polynomials of different rings. Thank you

 

[MindWalk]

> I found the remainder to be 4. If anyone has time, I believe I made a mistake somewhere and would like someone to attempt this division so I can check my work. If you have time please check it because I am not confident in dividing polynomials of different rings.

Since (x+3)(x^3 + 2x^2 + 4x + 1) = x^4 + 2x^3 + 4x^2 + x + 3x^3 + 6x^2 + 12x + 3 = x^4 + 5x^3 + 10x^2 + 13x + 3 = x^4 + 3x + 3 (mod 5), I deduce that the remainder is -1 = 4 (mod 5). What makes you think you (and therefore also I) made a mistake somewhere?