How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)^2 have?(1 point)
To determine the number of zeros, we need to consider the factors of the polynomial.
The function f(x)=x(x−1)(2x+4)^2 will have zeros at the values of x that make any of the factors equal to zero.
1. The factor x will be equal to zero when x=0.
2. The factor (x-1) will be equal to zero when x=1.
3. The factor (2x+4)^2 will be equal to zero when 2x+4=0. Solving for x, we get x=-2.
So, the function has three zeros: x=0, x=1, x=-2.
Therefore, the function f(x)=x(x−1)(2x+4)^2 has three zeros.
To determine the number of zeros, including repeated zeros, of the function f(x) = x(x - 1)(2x + 4)^2, we need to analyze the factors individually.
1. The factor x has one zero, which is x = 0.
2. The factor (x - 1) also has one zero, which is x = 1.
3. The factor (2x + 4) represents a linear equation. To find its zero, we set it equal to zero:
2x + 4 = 0
2x = -4
x = -2
Therefore, this factor has one zero at x = -2.
4. Since the factor (2x + 4) is squared, it counts as two zeros. So, altogether, this factor contributes two zeros.
Thus, the function f(x) = x(x - 1)(2x + 4)^2 has a total of 1 + 1 + 2 = 4 zeros, including repeated zeros.
To determine the number of zeros that the function has, we need to analyze each factor within the function equation individually.
First, let's consider the factor x. This factor will be zero when x = 0. Therefore, it contributes one zero to the function.
Next, let's look at the factor (x - 1). This factor will be zero when x = 1. Hence, it also contributes one zero to the function.
Now, let's focus on the term (2x + 4)^2. To find its zeros, we set (2x + 4) equal to zero:
2x + 4 = 0
Subtracting 4 from both sides gives:
2x = -4
Then we divide by 2:
x = -2
Therefore, the factor (2x + 4)^2 will be zero when x = -2. Since this factor is squared, it contributes two zeros to the function.
Overall, the function f(x) = x(x - 1)(2x + 4)^2 has a total of 4 zeros, including repeated zeros.