Question

In: Advanced Math

1. Solve the equations 256? ≡ 442(??? 60), 3? + 4 ≡ 6(??? 13). 5. Prove...

1. Solve the equations 256? ≡ 442(??? 60), 3? + 4 ≡ 6(??? 13).

5. Prove that ?2 + ? + 1 is an irreducible polynomial of degree 2.

Solutions

Expert Solution

1. Given equation  

, dividing by 2 throughout .

Now this equation have no solution as is always even and so is odd . Since even does not divides odd so 30 does not divides and so the equation have no solution .

Hence the equation have no solution .

Given equation ,

Multiplying both side by 4 we get ,

Hence the required solution are that is  

5. Given polynomial .

A second degree polynomial is reducible if and only if it has a root .

Now discrement of the polynomial is ,

As discreminant is negative so the polynomail has no real root and hence it is irreducible .

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If you have doubt or need more clarification at any step please comment .

Colby Messinger answered 6 months ago
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