How do you use the factor theorem to determine whether x+1 is a factor of # x^3 + x^2 + x + 1#?

1 Answer
Jan 3, 2016

The factor theorem states that a polynomial #f(x)# has a factor #(x+k)# if and only if #f(-k)=0#.
Here #x^3+x^2+x+1# is a polynomial.
Let #f(x)=x^3+x^2+x+1#

Now we want to know that is #x+1# a factor of #f(x)# or not.

For this purpose we have to put# x=-1# in #f(x)#, if the result comes to be #0# then #x+1# is a factor of #f(x)# and if the result comes not to be #0# then #x+1# is not a factor of #f(x)#.

Put #x=-1# in #f(x)#
#implies f(-1)=(-1)^3+(-1)^2+(-1)+1=-1+1-1+1=0#

Since the result is #0#, therefore #x+1# is a factor of the given polynomial.