How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)^2 have?(1 point)
The degree cannot be determined from the information provided.
4
3
2
To determine the number of zeros of a function, we need to look at the factors of the function.
Given: f(x) = x(x-1)(2x+4)^2
Let's go through each factor of the function:
1) x - This factor will be equal to zero when x = 0. So, there is one zero.
2) x - 1 - This factor will be equal to zero when x = 1. So, there is one more zero.
3) (2x + 4)^2 - This factor will be equal to zero when 2x + 4 = 0. Solving this equation gives x = -2. Therefore, there is one more zero.
In total, the function f(x) has 3 zeros, including repeated zeros. Therefore, the correct answer is 3.
To find the number of zeros for the function, we need to determine the number of times the function crosses the x-axis or "hits" zero.
The function f(x) = x(x - 1)(2x + 4)^2 will have zeros at three different values of x:
1. When x = 0 (from the factor x)
2. When x = 1 (from the factor x - 1)
3. When (2x + 4)^2 = 0
Simplifying, we have:
(2x + 4)^2 = 0
2x + 4 = 0 or -2x - 4 = 0
2x = -4 or -2x = 4
x = -2 or x = -2
So, the function has a repeated zero at x = -2 and zeros at x = 0 and x = 1.
Therefore, the function f(x) = x(x - 1)(2x + 4)^2 has 3 zeros. Answer: \boxed{3}.