Final answer:
The function f(x)=x(x-1)(2x+4)² has three zeros: at x=0, x=1, and a repeated zero at x=-2 which counts twice due to the square of that factor.
Step-by-step explanation:
To determine how many zeros the function f(x)=x(x−1)(2x+4)² has, we need to set the function equal to zero and solve for x. Doing so allows us to find the points where the function intersects the x-axis, which correspond to the zeros of the function.
- Factor Zero: When x=0, the first factor x equals zero.
- Factor x-1 Zero: When x=1, the second factor (x-1) equals zero.
- Factor (2x+4)² Zero: The third factor (2x+4)² equals zero when x=-2. Since it is squared, this zero is repeated.
Therefore, the function has three zeros: one at x=0, another at x=1, and a repeated zero at x=-2. Note that the repeated zero at x=-2 counts as two zeros because of the square on the factor.