I had a problem that states: "Determine all values of k such that x^2-kx-6 can be factored by x-2." I had no idea what the answer was and left it blank. Can you please tell me how i would go through doing a problem like that. Thank you.

2. Originally Posted by aussiekid90
I had a problem that states: "Determine all values of k such that x^2-kx-6 can be factored by x-2." I had no idea what the answer was and left it blank. Can you please tell me how i would go through doing a problem like that. Thank you.
Remember by the theory of polynomial $\displaystyle (x-2)$ is a factor of $\displaystyle f(x)=x^2-kx-6$ only if 2 is a zero of this polynomial, thus, $\displaystyle f(2)=0$, thus, $\displaystyle 4-2k-6=0$ thus, $\displaystyle k=-1$.

Now for the check, the polynomial is $\displaystyle x^2+x-6$ which factors into $\displaystyle (x+3)(x-2)$ thus, it works.
Q.E.D.

3. Would this be the only possiblity because the problem states find all possible solutions for K

4. Originally Posted by aussiekid90
Would this be the only possiblity because the problem states find all possible solutions for K
Yes, look at my derivation, only one possibility.