10. Find a polynomial with zeros $$ -2,3 $$, and 5 and passes through the point $$ (2,24) $$

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To find a polynomial with the given zeros and that passes through a specific point, we can follow these steps:

1. **Identify the zeros**: The zeros of the polynomial are $$ -2, 3, $$ and $$ 5 $$. This means that the polynomial can be expressed in factored form as:
   $$
   P(x) = k(x + 2)(x - 3)(x - 5)
   $$
   where $$ k $$ is a constant that we need to determine.

2. **Use the point to find $$ k $$**: We know that the polynomial passes through the point $$ (2, 24) $$. This means that when $$ x = 2 $$, $$ P(2) = 24 $$. We can substitute $$ x = 2 $$ into the polynomial and set it equal to 24:
   $$
   P(2) = k(2 + 2)(2 - 3)(2 - 5) = 24
   $$

3. **Calculate $$ P(2) $$**:
   $$
   P(2) = k(4)(-1)(-3) = k(12)
   $$
   So we have:
   $$
   12k = 24
   $$

4. **Solve for $$ k $$**:
   $$
   k = \frac{24}{12} = 2
   $$

5. **Write the final polynomial**: Now that we have $$ k $$, we can write the polynomial:
   $$
   P(x) = 2(x + 2)(x - 3)(x - 5)
   $$

6. **Expand the polynomial**: We can expand this expression to get the polynomial in standard form.

Let's perform the expansion step by step:

First, expand $$ (x + 2)(x - 3) $$:
$$
(x + 2)(x - 3) = x^2 - 3x + 2x - 6 = x^2 - x - 6
$$

Next, multiply this result by $$ (x - 5) $$:
$$
(x^2 - x - 6)(x - 5) = x^3 - 5x^2 - x^2 + 5x - 6x + 30 = x^3 - 6x^2 - x + 30
$$

Finally, multiply by $$ 2 $$:
$$
P(x) = 2(x^3 - 6x^2 - x + 30) = 2x^3 - 12x^2 - 2x + 60
$$

Thus, the polynomial is:
$$
\boxed{2x^3 - 12x^2 - 2x + 60}
$$
The polynomial is $$ 2x^3 - 12x^2 - 2x + 60 $$.

10. Find a polynomial with zeros \( -2,3 \), and 5 and passes through the point \( (2,24) \)

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