i)f(x)=(x^3 + -x^2 )(x^2 + 2)
To find the roots of the given function f(x), we need to find the values of x for which f(x) equals zero. This means we are looking for the x-values that make the expression (x^3 - x^2)(x^2 + 2) equal to zero.
To solve this, we set each factor of the expression equal to zero and solve for x individually.
First, let's set the factor (x^3 - x^2) equal to zero:
x^3 - x^2 = 0
Next, let's set the factor (x^2 + 2) equal to zero:
x^2 + 2 = 0
Solving the first equation, x^3 - x^2 = 0:
Factor out an x^2:
x^2(x - 1) = 0
This equation gives us two possibilities:
x^2 = 0 --> x = 0
or
x - 1 = 0 --> x = 1
Now let's solve the second equation, x^2 + 2 = 0:
Subtract 2 from both sides:
x^2 = -2
Since we have a negative value on the right side, there are no real solutions to this equation. However, there is a solution in the complex number system, which is represented as:
x = √(-2)
So, the roots of the function f(x) = (x^3 - x^2)(x^2 + 2) are x = 0, x = 1, and x = √(-2) (in the complex number system).