Final answer:
To find the values of k and c, we can set up a system of equations using the fact that (m+1) and (m-3) are factors of m2−km+c. Solving this system of equations, we find k = 0 and c = 0.
Step-by-step explanation:
To find the values of k and c, we need to use the fact that (m+1) and (m-3) are factors of m2−km+c. This means that when we substitute m = -1 and m = 3 into the equation, the equation should equal zero. Let's set up the equation:
(-1+1)(-1-3) = 0
(3+1)(3-3) = 0
Expanding these equations, we get:
0 = -4k + c
0 = 16k + c
Now, we have a system of equations. Solving these equations will give us the values of k and c:
Adding the two equations together, we get:
0 = 12k + 2c
Dividing by 2, we get:
0 = 6k + c
Substituting this equation into one of the previous equations, we get:
0 = -4k + (6k + c)
Simplifying, we get:
0 = 2k + c
From this equation, we can see that k = -c/2.
If we substitute this value of k into one of the original equations, we can solve for c:
0 = 6(-c/2) + c
Simplifying, we get:
0 = -3c + 2c
Solving for c, we get:
c = 0
Substituting this value of c into the equation k = -c/2, we get:
k = 0
Therefore, the values of k and c are k = 0 and c = 0.