Question

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

Answer 1

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.