Question

Refer to the polynomial function h(x) = -3x^2(5x+3)(x-2 that you wrote in Question 1 of Finding the Zeros of a Polynomial Function. Describe and provide an explanation for the end behavior of h(x).

Answer 1

To analyze the end behavior of the polynomial function \( h(x) = -3x^2(5x + 3)(x - 2) \), we first need to determine the degree and leading coefficient of the polynomial.

### Step 1: Find the degree of the polynomial
The function can be expressed as the product of its factors. We have:

1. The term \( -3x^2 \), which has a degree of 2.
2. The term \( (5x + 3) \), which has a degree of 1.
3. The term \( (x - 2) \), which has a degree of 1.

To find the total degree of \( h(x) \), we add the degrees of all terms:

\[
\text{Degree} = 2 + 1 + 1 = 4
\]

### Step 2: Identify the leading coefficient
The leading term of the polynomial comes from multiplying the leading coefficients of each factor. The leading term from \( -3x^2 \) is \( -3x^2 \), and the leading term from the polynomial expression \( (5x + 3)(x - 2) \) is \( 5x^2 \) (since \( 5x \cdot x \)). Thus, the leading term of \( h(x) \) is:

\[
-3x^2 \cdot 5x^2 = -15x^4
\]

So, the leading coefficient is \( -15 \).

### Step 3: Determine end behavior based on degree and leading coefficient
Now that we have established that \( h(x) \) is a polynomial of degree 4 (even degree) with a leading coefficient of \( -15 \) (negative), we can analyze the end behavior:

- For polynomials with an **even degree**:
- As \( x \) approaches \( +\infty \) (right end), \( h(x) \) approaches \( -\infty \) (downward).
- As \( x \) approaches \( -\infty \) (left end), \( h(x) \) also approaches \( -\infty \) (downward).

### Conclusion
The end behavior of the polynomial function \( h(x) = -3x^2(5x + 3)(x - 2) \) is described as follows:
- As \( x \to +\infty, h(x) \to -\infty \)
- As \( x \to -\infty, h(x) \to -\infty \)

This means that both ends of the graph of \( h(x) \) will trend downwards as you move to the left or right.