Question

Choose whole numbers to fill in the two blanks in the following polynomial function:
h(x) = -3x^2(5x +__) (x -___)

Answer 1

Find the zeros of h(x)

Answer 2

To fill in the blanks in the polynomial function \( h(x) = -3x^2(5x + \_\_) (x - \_\_) \), we can choose numbers that will maintain the polynomial's structure and form.

Let's analyze the structure:

1. The term \( 5x + \_\_ \): We need to choose a whole number that will be added to \( 5x \).
2. The term \( x - \_\_ \): We need to choose a whole number such that when this term is multiplied by \( 5x + \_\_ \), it results in an appropriately factored polynomial.

### Example Choices

Let's say we choose \( 2 \) for the first blank and \( 3 \) for the second blank. Thus, we have:

- \( 5x + 2 \)
- \( x - 3 \)

Now our polynomial function will look like this:

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

### Verification of Choices

You could expand it further if you want to verify:
1. First multiply the binomials:
\[
(5x + 2)(x - 3) = 5x^2 - 15x + 2x - 6 = 5x^2 - 13x - 6
\]

2. Then multiply by \(-3x^2\):
\[
h(x) = -3x^2(5x^2 - 13x - 6) = -15x^4 + 39x^3 + 18x^2
\]

So, the choice works correctly. Thus, one set of results for the blanks is:

- First blank: \( 2 \)
- Second blank: \( 3 \)

You could choose other whole numbers as long as they satisfy your requirements.

Answer 3

To find the zeros of the polynomial function

\[
h(x) = -3x^2(5x + 2)(x - 3),
\]

we need to set \( h(x) = 0 \) and solve for \( x \).

### Step 1: Set the function to zero

\[
-3x^2(5x + 2)(x - 3) = 0.
\]

### Step 2: Use the zero product property

The zero product property tells us that if a product of multiple factors equals zero, at least one of the factors must be equal to zero. So, we can set each factor to zero:

1. \( -3x^2 = 0 \)
2. \( 5x + 2 = 0 \)
3. \( x - 3 = 0 \)

### Step 3: Solve each equation

1. **For \( -3x^2 = 0 \)**:
\[
x^2 = 0 \quad \Rightarrow \quad x = 0.
\]

2. **For \( 5x + 2 = 0 \)**:
\[
5x = -2 \quad \Rightarrow \quad x = -\frac{2}{5}.
\]

3. **For \( x - 3 = 0 \)**:
\[
x = 3.
\]

### Step 4: List the zeros

The zeros of the function \( h(x) \) are:

- \( x = 0 \)
- \( x = -\frac{2}{5} \)
- \( x = 3 \)

Thus, the zeros of \( h(x) \) are \(\{0, -\frac{2}{5}, 3\}\).