a, \(x^4+x^3+x+1\)
\(=x^3\left(x+1\right)+\left(x+1\right)\)
\(=\left(x^3+1\right)\left(x+1\right)\)
\(=\left(x+1\right)^2\left(x^2-x+1\right)\)
b, \(x^2y+xy^2-x-y\)
\(=xy\left(x+y\right)-\left(x+y\right)=\left(xy-1\right)\left(x+y\right)\)
c, \(5xy^2-10xyz+5xz^2\)
\(=5x\left(y^2-2yz+z^2\right)\)
\(=5x\left(y-z\right)^2\)
e, \(x^4+x^2+1\)
\(=x^4+2x^2+1-x^2\)
\(=\left(x^2+1\right)-x^2\)
\(=\left(x^2-x+1\right)\left(x^2+x+1\right)\)
a) x\(^4\)+ x\(^3\)+ x + 1
\(=\) x\(^3\).( x + 1 ) + ( x + 1 )
\(=\) ( x\(^3\)+ 1 ).( x + 1 )
\(=\) ( x + 1 )\(^2\)( x\(^2\)- x + 1 )
b) x\(^2\).y + x.y\(^2\)- x - y
\(=\) x.y ( x + y ) - ( x + y )
\(=\) ( x.y - 1 ).( x + y )
c) 5xy\(^2\)- 10xyz + 5xz\(^2\)
\(=\) 5x( y\(^2\)- 2yz + z\(^2\))
\(=\) 5x( y - z )\(^2\)
d) x\(^2\)- x + 6
\(=\) x + 6
e) x\(^4\)+ x\(^2\)+ 1
\(=\) x\(^4\)+ 2x\(^2\)+ 1 - x\(^2\)
\(=\) ( x\(^2\)+ 1 ) - x\(^2\)
\(=\) ( x\(^2\)- x + 1 ).( x\(^2\)+ x + 1 )
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