Write the equation of a function with zeroes at $$ x=-2, x=1, x=2 $$. $$ f(x)=\square $$

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To write the equation of a function with zeroes at $$ x = -2 $$, $$ x = 1 $$, and $$ x = 2 $$, we can use the fact that if a function has zeroes at certain values, it can be expressed in factored form.

The zeroes indicate that the function can be written as:

$$
f(x) = k(x + 2)(x - 1)(x - 2)
$$

where $$ k $$ is a constant that can be any non-zero value. For simplicity, we can choose $$ k = 1 $$. Thus, the function becomes:

$$
f(x) = (x + 2)(x - 1)(x - 2)
$$

Now, we can expand this expression:

1. First, expand $$ (x - 1)(x - 2) $$:

$$
(x - 1)(x - 2) = x^2 - 2x - x + 2 = x^2 - 3x + 2
$$

2. Now, multiply this result by $$ (x + 2) $$:

$$
f(x) = (x + 2)(x^2 - 3x + 2)
$$

Expanding this:

$$
f(x) = x(x^2 - 3x + 2) + 2(x^2 - 3x + 2)
$$
$$
= x^3 - 3x^2 + 2x + 2x^2 - 6x + 4
$$
$$
= x^3 - 3x^2 + 2x + 2x^2 - 6x + 4
$$
$$
= x^3 - x^2 - 4x + 4
$$

Thus, the final equation of the function is:

$$
f(x) = x^3 - x^2 - 4x + 4
$$

So, the function can be expressed as:

$$
f(x) = (x + 2)(x - 1)(x - 2)
$$

or in expanded form:

$$
f(x) = x^3 - x^2 - 4x + 4
$$
$$ f(x) = (x + 2)(x - 1)(x - 2) $$ or $$ f(x) = x^3 - x^2 - 4x + 4 $$

Write the equation of a function with zeroes at \( x=-2, x=1, x=2 \). \( f(x)=\square \)

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