The factor theorem is basically a rule that states that if #p(a)=0# then your remainder is equal to 0 ie #R=0# and so #x-a=0#
What this means is if you say have a polynomial of #p(x)=x^2+2x+1# and you want to see whether #x=-1# is a factor or not.
You would sub in #x=-1# into the equation ie
#p(-1)=(-1)^2+2(-1)+1#
#p(-1)=1-2+1#
#p(-1)=0#
Now, notice that when you subbed in #x=-1#, your #p(-1)=0#. This means that your #x=-1# is a factor of the polynomial because your remainder is equal to 0 so #x-(-1)=x+1=0#
Now say you wanted to sub in #x=1# into the polynomial. ie
#p(1)=1^2+2(1)+1#
#p(1)=1+2+1#
#p(1)=4#
Now notice here that when you subbed in #x=1#, your #p(1) !=0# and instead it equals to 4. This means that your remainder is equal to 4 so #x=1# is NOT a factor of the polynomial so #x-1=0# is NOT a factor.
Say you have another polynomial #p(x)=x^3-8# and you wanted to see if #x=2# is a factor
You would sub #x=2# into the polynomial
#p(2)=2^3-8#
#p(2)=8-8#
#p(2)=0#
Since your remainder is equal to 0 ie #r=0#, then #x=2# is a factor of the polynomial
If you wanted to sub in #x=-2# into the polynomial
#p(-2)=(-2)^3-8#
#p(-2)=-8-8#
#p(-2)=-16#
Since your remainder is not equal to 0 ie #r=-16#, then #x=-2# is NOT a factor of the polynomial
