Refer to the polynomial function h(x) = 0 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

For the polynomial function \( h(x) = 0 \), the polynomial is a constant function. Its characteristics and end behavior can be analyzed as follows:

Description of End Behavior

1. Constant Value: The value of \( h(x) \) is always 0 for any value of \( x \). Therefore, regardless of how far \( x \) increases or decreases, \( h(x) \) remains constant at 0.
2. End Behavior: As \( x \) approaches positive infinity (\( x \to +\infty \)), \( h(x) \) remains 0.
3. Negative Infinity: Similarly, as \( x \) approaches negative infinity (\( x \to -\infty \)), \( h(x) \) again remains 0.

Explanation

• Graph Representation: If you were to graph \( h(x) = 0 \), it would be a horizontal line that spans the width of the graph at the height of 0 on the y-axis.
• Polynomial Characteristics: Since \( h(x) \) is a polynomial of degree 0 (as it is constant), there are no leading terms that would influence the growth behavior of the polynomial as \( x \) approaches either infinity.
• Behavior at Infinity: Therefore, the end behavior of \( h(x) \) can be summarized as: \[ \lim_{x \to +\infty} h(x) = 0 \quad \text{and} \quad \lim_{x \to -\infty} h(x) = 0 \]

In conclusion, the end behavior of the polynomial function \( h(x) = 0 \) is flat across all \( x \)—it remains at the constant value of 0 at both ends as \( x \) approaches positive and negative infinity.