how do u solve x^3+2x^2-6x-6=(x-2)(x+3)(x+k)
To solve the equation x^3 + 2x^2 - 6x - 6 = (x - 2)(x + 3)(x + k), we need to find the value of k.
To do that, we can follow the steps below:
Step 1: Expand the right side of the equation by multiplying (x - 2)(x + 3)(x + k):
(x - 2)(x + 3)(x + k) = (x^2 - 2x + 3x - 6)(x + k)
= (x^2 + x - 6)(x + k)
= x^3 + kx^2 + x^2 + kx - 6x - 6k
= x^3 + (k + 1)x^2 + (k - 6)x - 6k
Step 2: Compare the expanded expression from step 1 with the original equation:
x^3 + 2x^2 - 6x - 6 = x^3 + (k + 1)x^2 + (k - 6)x - 6k
Step 3: Equate the corresponding coefficients:
For the coefficients of x^2: 2 = k + 1 ---> k = 2 - 1 ---> k = 1
For the coefficients of x: -6 = k - 6 ---> k = -6 + 6 ---> k = 0
For the constant terms: -6 = -6k ---> k = -6 / (-6) ---> k = 1
Step 4: Check the value of k by substituting it back into the original equation:
x^3 + 2x^2 - 6x - 6 = (x - 2)(x + 3)(x + 1)
Therefore, k = 1 is the value that satisfies the equation.
So, the solution to the equation x^3 + 2x^2 - 6x - 6 = (x - 2)(x + 3)(x + k) is k = 1.