Answer:
The values are k = -2 and c = 1
Step-by-step explanation:
Since it's a division of polynomials we can calculate "c" and "k" by using the inverse operation, which is the product. We need to multiply "x + k" with "3x + 1" and sum it with 3, that should be equal to "3x² - 5x + c". We have:
[tex]3x^2 - 5x + c = (x + k)*(3x + 1) + 3\\3x^2 - 5x + c = 3x^2 + x + 3kx + k + 3\\3x^2 - 5x + c = 3x^2 + (1 + 3k)x + (k+3)[/tex]
In order for two polynomials to be equal, every coefficient must be equal, therefore:
[tex]-5 = 1 + 3k\\1 + 3k = -5\\3k = -5 -1\\3k = -6\\k = -2[/tex]
[tex]c = k + 3\\c = -2 + 3\\c = 1[/tex]
