Determine whether the following polynomial has (x+1) as a factor:
x4−x3+x2−x+1.
To check whether (x+1) is the factor of polynomial x4−x3+x2−x+1:
We take, x+1=0 ⟹ x=−1.
Replace x by −1, we get,
p(x)=x4−x3+x2−x+1
p(−1)=(−1)4−(−1)3+(−1)2−(−1)+1
p(−1)=1−(−1)+1−(−1)+1=1+1+1+1+1
p(−1)=5.
That is, value of p(−1) is not zero.
Then x+1 is not the factor of polynomial x4−x3+x2−x+1.