If (x2 + x+ 1 ) is a factor of the polynomial 3x3 + 8x2 + 8x + 3 + 5k. find the value of k

One way to do this is apply the polynomial remainder theorem. First, note that



which means either of are roots to .

The polynomial remainder theorem states that the remainder upon dividing a polynomial by a linear expression is equal to the value of .

So what our factorization above tells us is that if we plug in, for example, , we would get



Expanding the left hand side gives



Alternatively, you could have determined the quotient and remainder via long division, which you would find to be



Then, since you know is a factor of the cubic polynomial, you also know that the remainder should be 0, which means we're left with the equation



as expected.