Let p(x) = polynomial anxn+an-1xn-1+...a1x+a0 if an>0 and x is even then which of the following statements is true? 1) x➡️ infinity then f(x)➡️ negative infinity 2) x➡️ negative infinity then f(x)➡️ negative infinity 

Given the polynomial \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0 \) with \( a_n > 0 \) and \( x \) even, let's analyze the behavior of \( p(x) \) as \( x \to \infty \) and \( x \to -\infty \).

1. \( \lim_{x \to \infty} p(x) \)
2. \( \lim_{x \to -\infty} p(x) \)

Since \( x \) is even, consider \( x = 2k \), where \( k \in \mathbb{Z} \).

### Analysis for \( x \to \infty \):
When \( x \to \infty \), we focus on the leading term \( a_n x^n \) because it dominates the behavior of the polynomial. Given \( a_n > 0 \):

\[ a_n x^n \to \infty \]

So, \( p(x) \to \infty \) as \( x \to \infty \).

### Analysis for \( x \to -\infty \):
When \( x \to -\infty \), and since \( n \) is an integer and \( x \) is even, the term \( x^n \) behaves as:

\[ x^n = (-|x|)^n = |x|^n \]

This means that even if \( x \to -\infty \), \( x^n \) remains positive. Thus,

\[ a_n x^n \to \infty \]

Hence, \( p(x) \to \infty \) as \( x \to -\infty \).

Based on these analyses:
- Statement 1: \( x \to \infty \) then \( p(x) \to -\infty \) is **false**.
- Statement 2: \( x \to -\infty \) then \( p(x) \to -\infty \) is **false**.

Thus, neither statement is true.

Do you want details or have any questions? Here are some questions you might ask next:

1. Can you explain the behavior of polynomials with odd degree terms?
2. What happens if the leading coefficient \( a_n \) is negative?
3. How does the parity (odd/even) of the highest degree term affect the polynomial's behavior at infinity?
4. What is the significance of the sign of the leading coefficient in polynomials?
5. Can we analyze the behavior of polynomial functions with complex coefficients?

**Tip:** Always consider the leading term's degree and coefficient when analyzing polynomial behavior for large values of \( x \).

Explore the behavior of polynomial functions with even exponents as x approaches infinity and negative infinity. Understand why statements regarding the limit of these functions are false. Suitable for advanced high school students studying polynomial functions.

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