If  is a factor of , what is the value of k?

1. The remainder theorem tells us that you get the same thing when you
substitute 3 into a polynomial as you get when you divide the polynomial
by x-3 and take only the remainder.

2. The factor theorem thell us that since x-3 is a factor of the polynomial
then if we divided the polynomial by x-3, the remainder would be 0.

Putting these two facts together we can see that if we substituted 3 into
the polynomial, we will get the same result as the remainder would be if we
divided the polynomial by x-3. And furthermore due to 2, that remainder must
be 0.  So all we have to do is substitute 3 for x in the polynomial and set 
it equal to 0.

So substituting 3 for x in x^4-3x^3+kx+3 gives

                           3^4-3(3)^3+k(3)+3
                          
                             81-3(27)+3k+3
                          
                              81-81+3k+3

                                 3k+3

Setting 3k+3 = 0
          3k = -3
           k = -1

Now let's check to see if we are right. If we are then the polynomial

 becomes . We will divide that 

synthetically by x-3 to see if we get a 0 remainder.  First we must
insert a + term, and write it as .  

      3 | 1 -3  0 -1  3
        |    3  0  0 -3  
          1  0  0 -1  0     
  
Sure enough, we do get 0 for a remainder.
So k = -1 is correct.

Edwin