How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)^2 have?(1 point)
1. The degree cannot be determined from the information provided.
2. 4
3. 3
4. 2
To find the zeros of the function, we need to set it equal to zero and solve for x:
x(x-1)(2x+4)^2 = 0
Since we have a product of factors equal to zero, one or more of the factors must be equal to zero.
Setting x = 0:
(0)(0-1)(2(0)+4)^2 = 0
This simplifies to:
0 * (-1) * 4^2 = 0
0 * -1 * 16 = 0
0 = 0
So one zero is x = 0.
Setting x - 1 = 0:
(x-1)(0)(2x+4)^2 = 0
This simplifies to:
(x-1) * 0 * 4^2 = 0
0 * 0 * 16 = 0
0 = 0
So another zero is x = 1.
Setting 2x + 4 = 0:
x(x-1)(0)^2 = 0
This simplifies to:
x(x-1) * 0 = 0
0 * 0 = 0
0 = 0
Since this equation is always true, the factor 2x + 4 does not contribute any additional zeros.
Therefore, the function f(x) = x(x-1)(2x+4)^2 has 2 zeros: x = 0 and x = 1.
The answer is 4.
To determine the number of zeros, we need to analyze the factors of the function f(x).
The function f(x) has three factors: x, (x-1), and (2x+4)^2.
The factor x implies that the function has a zero at x=0.
The factor (x-1) implies that the function has a zero at x=1.
The factor (2x+4)^2 implies that the function has two zeros: one at x=-2 and one at x=-2.
So, in total, the function f(x) has 1 + 1 + 2 = 4 zeros, including repeated zeros.
Therefore, the correct answer is 2.
To determine the number of zeros, including repeated zeros, of the function f(x)=x(x−1)(2x+4)^2, we need to identify the values of x that make the function equal to zero.
To find the zeros of the function, we set f(x) equal to zero and solve for x.
Setting f(x) to zero:
x(x−1)(2x+4)^2 = 0
Now, we can analyze each factor separately to see when it would be equal to zero.
Factor 1: x
For this factor to be equal to zero, x must be equal to zero.
Factor 2: (x−1)
For this factor to be equal to zero, x must be equal to 1.
Factor 3: (2x+4)^2
To find the zeros of this factor, we need to solve for x in the equation (2x+4)^2 = 0.
Taking the square root of both sides, we get:
2x+4 = 0
Solving for x:
2x = -4
x = -2
So, the zeros of the function are x = 0, x = 1, and x = -2.
However, the question asks for the number of zeros, including repeated zeros. Since the factor (2x+4)^2 is squared, it will count as two zeros. Therefore, there are a total of 3 zeros, including the repeated zero.
The correct answer is option 3. 3.