Question

Find a polynomial equation that has zeros at x = - 2 x = O x = 3 and x = 5

y = x(x + 2)(x - 3)(x - 5)

y = (x + 2)(x - 3)(x - 5)

y = (x - 2)(x - 3)(x - 5)

y = (x - 2)(x + 3)(x + 5)

Answer 1

To find a polynomial that has zeros at \( x = -2 \), \( x = 0 \), \( x = 3 \), and \( x = 5 \), we can write it in factored form based on its zeros:

The factors corresponding to the given zeros are:
- \( (x + 2) \) for the zero at \( x = -2 \)
- \( (x - 0) \) or simply \( x \) for the zero at \( x = 0 \)
- \( (x - 3) \) for the zero at \( x = 3 \)
- \( (x - 5) \) for the zero at \( x = 5 \)

Therefore, the polynomial can be written as:

\[
y = (x + 2)(x)(x - 3)(x - 5)
\]

Now, let's analyze the answer choices provided:

1. \( y = x(x + 2)(x - 3)(x - 5) \) - This includes all zeros, which includes the correct factors.
2. \( y = (x + 2)(x - 3)(x - 5) \) - This does not include the zero at \( x = 0 \).
3. \( y = (x - 2)(x - 3)(x - 5) \) - This does not have any of the correct zeros specified.
4. \( y = (x - 2)(x + 3)(x + 5) \) - This also does not match any of the specified zeros.

Among the choices, the only polynomial that correctly represents the given zeros is:

\[
\boxed{y = x(x + 2)(x - 3)(x - 5)}
\]