How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)2 have?(1 point)
Responses
The degree cannot be determined from the information provided.
The degree cannot be determined from the information provided.
2
2
4
4
3
The function f(x) has a degree of 4 because the highest power of x is 4 in the expression (2x+4)^2.
To find the number of zeros, we need to look at the factors of the function. In this case, the function is f(x) = x(x-1)(2x+4)^2.
First, let's look at the factor x. We can see that x = 0 is a zero of the function, since when x = 0, the entire term x becomes zero.
Next, let's look at the factor (x-1). We can see that x = 1 is another zero of the function, since when x = 1, the entire term (x-1) becomes zero.
Finally, let's look at the factor (2x+4)^2. We notice that (2x+4)^2 is always positive, which means it will never equal zero. Therefore, this factor does not contribute any additional zeros to the function.
Taking all of this into account, we have a total of 2 zeros for the function f(x)=x(x−1)(2x+4)^2. So the correct answer is:
2
To find the number of zeros of the function f(x), we need to solve the equation f(x) = 0. Let's break down the process step by step:
1. Start with the function f(x) = x(x − 1)(2x + 4)^2.
2. Set f(x) = 0 and solve the equation. In this case, we have:
x(x − 1)(2x + 4)^2 = 0.
3. To solve this equation, we set each factor equal to zero and solve for x:
x = 0 or x − 1 = 0 or (2x + 4)^2 = 0.
For the first factor, x = 0.
For the second factor, x = 1.
4. For the third factor, (2x + 4)^2 = 0, we apply the square root property:
2x + 4 = 0 or 2x + 4 = 0.
Solving these equations, we find:
2x = -4 or 2x = -4.
Dividing both sides by 2, we get:
x = -2 or x = -2.
5. Combining all the values obtained, we have the following zeros:
x = 0, 1, -2, -2.
So, the function f(x) = x(x − 1)(2x + 4)^2 has a total of 4 zeros, including repeated zeros.