Question

find all zeros of f(x)=x^4-4x^2

Answer 1

To find the zeros of the function \( f(x) = x^4 - 4x^2 \), we can start by factoring the expression.

1. First, notice that we can factor out \( x^2 \):
\[
f(x) = x^2 (x^2 - 4)
\]

2. Next, we can factor \( x^2 - 4 \) further, as it is a difference of squares:
\[
x^2 - 4 = (x - 2)(x + 2)
\]

3. Putting it all together, we can rewrite the function as:
\[
f(x) = x^2 (x - 2)(x + 2)
\]

Now we can find the zeros by setting each factor equal to zero:
- From \( x^2 = 0 \):
\[
x = 0
\]

- From \( x - 2 = 0 \):
\[
x = 2
\]

- From \( x + 2 = 0 \):
\[
x = -2
\]

Thus, the zeros of the function \( f(x) = x^4 - 4x^2 \) are:
\[
x = 0, \quad x = 2, \quad x = -2
\]

In summary, the complete list of zeros is:
\[
\boxed{0, 2, -2}
\]