https://file.aiquickdraw.com/mathbot/file/1734278198505hgvbx9pl.jpg

Let's analyze the question. The polynomial function is:

\[
p(x) = x^3 - 10x^2 - 4
\]

We are asked to determine the behavior of \( p(x) \) as the input values (\( x \)) **decrease without bound**, i.e., as \( x \to -\infty \).

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### Step 1: Focus on the leading term
In any polynomial function, the **leading term** (the term with the highest power of \( x \)) dominates the behavior as \( x \to \pm \infty \).

Here, the leading term is \( x^3 \), since it has the highest power.

- When \( x \to -\infty \), the leading term \( x^3 \to -\infty \) (because the cube of a negative number remains negative and gets larger in magnitude).

The other terms \( -10x^2 \) and \( -4 \) become insignificant compared to \( x^3 \) as \( x \to -\infty \).

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### Step 2: Overall behavior of \( p(x) \) as \( x \to -\infty \)
Since \( x^3 \to -\infty \) dominates, the entire function \( p(x) \) will also approach \( -\infty \).

Thus, the output values **decrease without bound**.

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### Correct Answer:
**A) The output values decrease without bound.**

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Let me know if you'd like a further breakdown or visualization!

### Related Questions:
1. How do we determine the behavior of a polynomial for large positive or negative inputs?
2. What role does the degree of a polynomial play in determining end behavior?
3. How would the function behave as \( x \to +\infty \)?
4. What happens if the leading coefficient of \( x^3 \) is negative?
5. How does adding or subtracting constant terms affect the end behavior of a polynomial?

### Tip:
The leading term of a polynomial determines its end behavior. Always focus on the highest power of \( x \) when analyzing limits at infinity.

The polynomial function p(x) = x^3 - 10x^2 - 4. Which of the following describes the behavior of p as the input values decrease without bound?

Discover how to determine the behavior of the polynomial function p(x) = x^3 - 10x^2 - 4 as the input values decrease without bound. Learn about end behavior, leading terms, and polynomial analysis, suitable for Grades 10-12.

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