Final answer:
To find the zeros of the function f(x) = (x-2)(x²+3x+2), it must be factored completely to reveal (x-2)(x+1)(x+2). Setting each factor equal to zero gives the zeros x = 2, x = -1, and x = -2.
Step-by-step explanation:
To find the zeros of the function f(x) = (x-2)(x²+3x+2), we need to factor the quadratic expression completely and then solve for x where f(x) equals 0.
The given quadratic expression is already partially factored. It is a product of a linear term (x-2) and a quadratic term (x²+3x+2). We need to factor the quadratic term further. The quadratic x²+3x+2 can be factored into (x+1)(x+2).
So, the fully factored form of f(x) is: f(x) = (x-2)(x+1)(x+2). The zeros are found by setting each factor equal to zero:
- x - 2 = 0 → x = 2
- x + 1 = 0 → x = -1
- x + 2 = 0 → x = -2
Therefore, the zeros of the function are x = 2, x = -1, and x = -2.