`P(x)=ax^(4)+x^(3)-bx^(2)-4x+c.` if P(x) increases without bound as x increases without bound, then, as x decreases without bound, P(x)

Options

A. increases without bound
B. decreases without bound
C. approaches zero from above the x-axis
D. approaches zero from below the x-axis

Explanation

To solve the problem, we need to analyze the polynomial \( P(x) = ax^4 + x^3 - bx^2 - 4x + c \) and determine its behavior as \( x \) decreases without bound, given that it increases without bound as \( x \) increases without bound.

### Step-by-Step Solution:

1. **Identify the Leading Term**:
The leading term of the polynomial \( P(x) \) is \( ax^4 \). The behavior of the polynomial as \( x \) approaches positive or negative infinity is primarily determined by this term.

**Hint**: Focus on the term with the highest degree when analyzing polynomial behavior at extremes.

2. **Behavior as \( x \to +\infty \)**:
Since it is given that \( P(x) \) increases without bound as \( x \) increases without bound, we can conclude that \( a > 0 \). This is because for \( ax^4 \) to dominate and lead to an increase, \( a \) must be positive.

**Hint**: Consider the sign of the leading coefficient to determine the direction of the polynomial's growth.

3. **Behavior as \( x \to -\infty \)**:
Now, we need to analyze \( P(x) \) as \( x \) decreases without bound (i.e., \( x \to -\infty \)). The leading term \( ax^4 \) will still dominate the polynomial since it is of the highest degree.

4. **Evaluate \( ax^4 \) as \( x \to -\infty \)**:
As \( x \) approaches negative infinity, \( x^4 \) remains positive (since any negative number raised to an even power is positive). Therefore, \( ax^4 \) will still be positive and will increase without bound as \( x \to -\infty \) if \( a > 0 \).

**Hint**: Remember that even powers of negative numbers yield positive results.

5. **Conclusion**:
Since \( a > 0 \) and \( ax^4 \) dominates the polynomial \( P(x) \) as \( x \to -\infty \), we conclude that \( P(x) \) also increases without bound as \( x \) decreases without bound.

**Final Answer**: As \( x \) decreases without bound, \( P(x) \) increases without bound.