Sure! Let's consider a specific polynomial \( p(x) \) and analyze it using concrete numbers. Suppose we have:

\[ p(x) = 2x^4 + 3x^3 - 5x^2 + 4x - 1 \]

Here, the highest degree term is \( 2x^4 \), and its coefficient \( a_4 = 2 \) is positive. We will analyze the behavior as \( x \to \infty \) and \( x \to -\infty \) with \( x \) being an even integer.

### Case 1: \( x \to \infty \)

When \( x \to \infty \), the term \( 2x^4 \) will dominate because it grows faster than any other term in the polynomial. Let's evaluate the polynomial for a large even value of \( x \):

Suppose \( x = 10 \):

\[ p(10) = 2(10)^4 + 3(10)^3 - 5(10)^2 + 4(10) - 1 \]

Calculating each term separately:

\[ 2(10)^4 = 2 \cdot 10000 = 20000 \]
\[ 3(10)^3 = 3 \cdot 1000 = 3000 \]
\[ -5(10)^2 = -5 \cdot 100 = -500 \]
\[ 4(10) = 40 \]
\[ -1 = -1 \]

Summing these values:

\[ p(10) = 20000 + 3000 - 500 + 40 - 1 = 22499 \]

As you can see, \( p(x) \) is a large positive number, and as \( x \) continues to increase, \( p(x) \) will grow larger and larger. Hence, \( \lim_{x \to \infty} p(x) = \infty \).

### Case 2: \( x \to -\infty \)

Now consider \( x \to -\infty \). Let's evaluate the polynomial for a large negative even value of \( x \):

Suppose \( x = -10 \):

\[ p(-10) = 2(-10)^4 + 3(-10)^3 - 5(-10)^2 + 4(-10) - 1 \]

Calculating each term separately:

\[ 2(-10)^4 = 2 \cdot 10000 = 20000 \]
\[ 3(-10)^3 = 3 \cdot (-1000) = -3000 \]
\[ -5(-10)^2 = -5 \cdot 100 = -500 \]
\[ 4(-10) = -40 \]
\[ -1 = -1 \]

Summing these values:

\[ p(-10) = 20000 - 3000 - 500 - 40 - 1 = 16459 \]

As you can see, \( p(x) \) is also a large positive number for large negative values of \( x \). Hence, \( \lim_{x \to -\infty} p(x) = \infty \).

### Conclusion

From the above calculations, we see that as \( x \to \infty \), \( p(x) \to \infty \), and as \( x \to -\infty \), \( p(x) \to \infty \). This confirms that:

- \( x \to \infty \) then \( p(x) \to -\infty \) is false.
- \( x \to -\infty \) then \( p(x) \to -\infty \) is false.

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

1. How does the leading coefficient being negative change the polynomial's behavior?
2. What if the highest degree term is odd, like \( x^3 \)?
3. How can we find the roots of this polynomial?
4. What are some real-world applications of polynomial functions?
5. Can you explain polynomial division?

**Tip:** When evaluating polynomials, focus on the term with the highest degree for large values of \( x \) since it dominates the polynomial's behavior.

Explore polynomial behavior with concrete examples and numerical analysis. Learn how to evaluate polynomials and analyze their behavior as x approaches infinity and negative infinity. Suitable for high school students studying algebra and calculus.

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