When the function q(x) = x3 + kx2 – 7x + 3 is divided by (x + 1) the remainder is seven times theremainder that is found when the function is divided by (x + 2).Find the value of k

To solve this problem, we will use the Remainder Theorem which states that the remainder of a polynomial f(x) divided by (x-a) is f(a).

1. According to the problem, when q(x) is divided by (x + 1), the remainder is 7 times the remainder when q(x) is divided by (x + 2). So, we can write this as: q(-1) = 7q(-2).

2. Substituting x = -1 and x = -2 into the function q(x) = x^3 + kx^2 - 7x + 3, we get:

   q(-1) = (-1)^3 + k(-1)^2 - 7(-1) + 3 = -1 + k + 7 + 3 = k + 9

   q(-2) = (-2)^3 + k(-2)^2 - 7(-2) + 3 = -8 + 4k + 14 + 3 = 4k + 9

3. Now, substitute q(-1) and q(-2) into the equation q(-1) = 7q(-2) to get:

   k + 9 = 7(4k + 9)

4. Simplify the equation to solve for k:

   k + 9 = 28k + 63

   27k = -54

   k = -54 / 27

   k = -2

So, the value of k is -2.