Show that 4x2+6x+3 is a unit in Z8[x].All that I know that I have to do is find a polynomial such that(4x2+6x+3)(polynomial)=1, but I'm not sure the steps in general to follow. I think after multiplying the two polynomials the coefficients will have to turn out to be 0 mod 8 and the constant term will have to be 1 mod 8.  How do I approach any problem of this type?  I have no examples to go by, and I need to know the thinking process that is needed to solve these types of problems.  Thanks!

Question
Asked Nov 1, 2019

Show that 4x2+6x+3 is a unit in Z8[x].

All that I know that I have to do is find a polynomial such that

(4x2+6x+3)(polynomial)=1, but I'm not sure the steps in general to follow. I think after multiplying the two polynomials the coefficients will have to turn out to be 0 mod 8 and the constant term will have to be 1 mod 8.  How do I approach any problem of this type?  I have no examples to go by, and I need to know the thinking process that is needed to solve these types of problems.  Thanks!

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Expert Answer

Step 1

Given polynomial is,

4x2+6x+3 in Z[x
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4x2+6x+3 in Z[x

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Step 2

Consider a polynomial as,

x
2x 3 in Z
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x 2x 3 in Z

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Step 3

Then the product of both t...

(4x 6x3(2.x+ 3) = 8x2 +12x2 +12x2 +18x +6x +9
= 8.x3 +24.x2 + 24x +9
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(4x 6x3(2.x+ 3) = 8x2 +12x2 +12x2 +18x +6x +9 = 8.x3 +24.x2 + 24x +9

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