How many zeros, including repeated zeros, does the function f(x)=x(x−1)(2x+4)^2 have?

Answer 1

To find the number of zeros of the function \( f(x) = x(x - 1)(2x + 4)^2 \), we need to analyze each factor of the function.

1. First Factor: \( x \)
   Setting \( x = 0 \) gives us one zero.

2. Second Factor: \( x - 1 \)
   Setting \( x - 1 = 0 \) gives us \( x = 1 \). This is another zero.

3. Third Factor: \( (2x + 4)^2 \)
   We first set \( 2x + 4 = 0 \) which gives us: \[ 2x = -4 \implies x = -2 \] Since this factor is squared, the zero at \( x = -2 \) counts as a repeated zero. Hence, we count it as 2 zeros.

Now let's summarize the zeros we found:

• From \( x = 0 \): 1 zero
• From \( x - 1 = 0 \): 1 zero
• From \( (2x + 4)^2 = 0 \): 2 zeros

Adding them together gives: \[ 1 + 1 + 2 = 4 \]

Therefore, the total number of zeros (including repeated zeros) of the function \( f(x) = x(x - 1)(2x + 4)^2 \) is \( \boxed{4} \).