To find the roots of the function f(x) = x³ + x² – 6x, we need to solve the equation f(x) = 0. This means we set the function equal to zero:
x³ + x² – 6x = 0
First, we notice that each term on the left side of the equation has a common factor of x. We can factor this out:
x(x² + x – 6) = 0
This gives us one root right away, which is x = 0. Now we need to find the roots of the quadratic expression x² + x – 6 by factoring or using the quadratic formula.
To factor x² + x – 6, we look for two numbers that multiply to -6 and add up to 1. The numbers 3 and -2 work because 3 * -2 = -6 and 3 + (-2) = 1.
Thus, we can rewrite the equation as:
(x + 3)(x – 2) = 0
Now, we can set each factor equal to zero to find the roots:
x + 3 = 0 or x – 2 = 0
This gives us two additional roots: x = -3 and x = 2.
In conclusion, the roots of the function f(x) = x³ + x² – 6x are:
- x = 0
- x = -3
- x = 2