a) \(x^5+x-1\)
\(=x^5-x^4+x^3+x^4-x^3+x^2-x^2+x-1\)
\(=x^3\left(x^2-x+1\right)+x^2\left(x^2-x+1\right)-\left(x^2-x+1\right)\)
\(=\left(x^2-x+1\right)\left(x^3+x^2-1\right)\)
b) \(x^5+x^4+1\)
\(=x^5-x^3+x^2+x^4-x^2+x+x^3-x+1\)
\(=x^2\left(x^3-x+1\right)+x\left(x^3-x+1\right)+\left(x^3-x+1\right)\)
\(=\left(x^3-x+1\right)\left(x^2+x+1\right)\)
1 .
x5 + x + 1
= x5 - x2 + x2 + x + 1
= x2( x3 - 1) + ( x2 + x + 1)
= x2( x - 1)( x2 + x + 1) + ( x2 + x + 1)
= ( x2 + x + 1)( x3 - x2 + 1)
2 .
x5 + x4 + 1
= x5 + x4 + x3 - x3 + 1
= x3( x2 + x + 1) - ( x3 - 1)
= x3( x2 + x + 1) - ( x - 1)( x2 + x + 1)
= ( x2 + x + 1)( x3 - x + 1)
3. x8 + x + 1
= x8 - x2 + x2 + x + 1
= x2( x6 - 1) + ( x2 + x + 1)
= x2( x3 - 1)( x3 + 1) + ( x2 + x + 1)
= ( x5 + x2)( x - 1)( x2 + x + 1) + ( x2 + x + 1)
= ( x2 + x + 1)( x6 - x5 + x3 - x2 + 1)
4. x8 + x7 + 1
= x8 + x7 + x6 - x6 + 1
= x6( x2 + x + 1) - [ ( x3)2 - 1 ]
= x6( x2 + x + 1) - ( x3 - 1)( x3 + 1)
= x6( x2 + x + 1) - ( x - 1)( x2 + x + 1)( x3 + 1)
= ( x2 + x +1 )[ x6 - ( x - 1)( x3 + 1) ]
= ( x2 + x +1 )( x6 - x4 - x + x3 + 1)
