first you want to find a.
see the attached worksheet.
step 1 shows the equation.
step 2 shows the result of synthetic division by a.
the remainder is a^2 + 6a + k
step 3 shows the result of synthetic division by 2a.
the remainder is 4a^2 + 12a + k
step 4 sets both these remainders equal to each other and then subtracts the expression on the left side of the equal sign from both sides of the equation to get 3a^2 + 6a = 0
step 5 factors the equation from step 4.
step 6 shows the solution.
a = 0 or a = -2
next you want to find k when the remainder is 2.
see the attached worksheet.
step 7 divides the equation by a = 0 and by 2a = 0.
since they are both = to 0, step 7 applies to both cases.
the result of the division is that the remainder = k when a or 2a = 0.
step 8 divides the equation by a = -2 and gets a remainder of k - 8.
step 9 divides the equatin by 2a = -4 and gets a remainder of k - 8.
both remainder are the same, as they should be.
if the remainder is 2, then k = 2 when a = 0 or 2a = 0.
step 10 makes k = 2 and divides by 0 to confirm that the remainder is 2.
step 11 makes k = 10 and divides by -2 to confirm that the remainder is 2.
division by -4 is not performed since the remainder is the same, so k = 10 applies to both cases.
your questions were:
When the polynomial f(x)=x^2 +6x +k is divided by x-a, the remainder is the same as when is divided by x-2a. find
a) the values of a,
b) the values of k if the remainder is 2.
answer to question a is that the values of a can be 0 or -2.
answer to question b is that, when the remainder is 2, the value of k is equal to 2 if a = 0, and the value of k is equal to 10 if a = -2.