One of the solutions to the equation x^3 + x^2 - 16x - 16 = 0 is -4
test the solution to find the full solution set
To test whether -4 is a solution to the equation x^3 + x^2 - 16x - 16 = 0, we substitute -4 into the equation and see if it satisfies the equation.
When we substitute x = -4 into the equation, we get:
(-4)^3 + (-4)^2 - 16(-4) - 16 = 0
-64 + 16 + 64 - 16 = 0
0 = 0
Since the equation is satisfied when x = -4, we have confirmed that -4 is indeed a solution.
To find the full solution set, we can use polynomial division or synthetic division to divide the equation by (x + 4). Doing so, we get:
(x^3 + x^2 - 16x - 16) ÷ (x + 4) = x^2 - 3x - 4
The quotient x^2 - 3x - 4 represents the remaining part of the equation after dividing by (x + 4). To find the solutions to x^2 - 3x - 4 = 0, we can factor or use the quadratic formula.
Factoring:
(x - 4)(x + 1) = 0
Setting each factor equal to zero and solving for x, we get:
x - 4 = 0 or x + 1 = 0
x = 4 or x = -1
Thus, the full solution set for the equation x^3 + x^2 - 16x - 16 = 0 is:
x = -4, 4, -1