Determine which of the following polynomials has (x+1) a factor:
(i) x³+x²+x+1
(ii) x⁴+x³+x²+x+1
(iii) x⁴+3x³+3x²+x+1
(iv) x³−x²−(2+√2)x+√2

Apply remainder theorem
x+1=0
x=-1
Put the value of x=-1 in all equations
(i) x³+x²+x+1=(-1)³+(-1)²+(-1)+1=-1+1-1+1=0
Then x+1 is the factor of the equation
(ii) x⁴+x³+x²+x+1=(-1)⁴+(-1)³+(-1)²+(-1)+1=1-1+1-1+1=1
This is not zero. Then x+1 is not the factor of the equation
(iii) x⁴+3x³+3x²+x+1=(-1)⁴+3(-1)³+3(-1)²+(-1)+1=1-3+3-1+1=1
This is not zero. Then x+1 is not the factor of equation
(iv) x³−x²−(2+√2)x+√2=(-1)³−(-1)²−(2+√2)(-1)+√2=-1-1+2+√2+√2=2√2
This is not zero. Then x+1 is not the factor of the equation.