How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)2 have?
To determine the number of zeros of a function, we need to examine the factors of the function.
The function $f(x)$ has zeros when any of its factors equal zero.
The first factor, $x$, equals zero when $x = 0$. So, $f(x)$ has one zero at $x=0$.
The second factor, $(x-1)$, equals zero when $x=1$. So, $f(x)$ has one zero at $x=1$.
The third factor, $(2x+4)$, equals zero when $x=-2$. So, $f(x)$ has one zero at $x=-2$.
Since $(2x+4)$ is squared, $f(x)$ will have an additional zero at $x=-2$ (a repeated zero).
Therefore, $f(x)$ has a total of $\boxed{4}$ zeros, including repeated zeros.
To find the number of zeros, we first need to factorize the equation.
f(x) = x(x - 1)(2x + 4)^2
Now, let's analyze each factor:
1. x: This factor will be zero when x = 0.
2. (x - 1): This factor will be zero when x = 1.
3. (2x + 4)^2: This factor will be zero when 2x + 4 = 0. Solving for x, we get x = -2.
Now, let's count the number of zeros:
1. x = 0: This is one zero.
2. x = 1: This is another zero.
3. x = -2: This is a repeated zero because the factor (2x + 4)^2 has a multiplicity of 2.
Hence, the function f(x) = x(x - 1)(2x + 4)^2 has a total of 3 zeros, including repeated zeros.
To determine the number of zeros of the function f(x) = x(x - 1)(2x + 4)^2, we can look at the factors individually.
1. Zero from x:
The factor x will be zero when x = 0. Therefore, there is one zero at x = 0.
2. Zero from (x - 1):
The factor (x - 1) will be zero when x = 1. Therefore, there is one zero at x = 1.
3. Zero from (2x + 4)^2:
The factor (2x + 4)^2 will be zero when 2x + 4 = 0. Solving this equation, we get:
2x + 4 = 0
2x = -4
x = -2
Since the squared term (2x + 4)^2 gives us a duplicate zero, we count this zero twice. Therefore, there are two zeros at x = -2.
Combining all the zeros, we have: x = 0, x = 1, and x = -2. So, the function f(x) = x(x - 1)(2x + 4)^2 has a total of three zeros.