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To understand the behavior of the polynomial [tex]\( f(x) = -5x^6 + 3x^4 - 2x^3 - 8 \)[/tex] as [tex]\( x \)[/tex] approaches positive and negative infinity, let's analyze each term and consider their effects on [tex]\( f(x) \)[/tex].
### Step-by-Step Analysis
1. Highest Degree Term Dominance:
The term with the highest degree in the polynomial will dominate the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches positive or negative infinity. In this polynomial, the highest degree term is [tex]\( -5x^6 \)[/tex].
2. Behavior as [tex]\( x \)[/tex] Approaches Positive Infinity:
When [tex]\( x \)[/tex] becomes a very large positive number:
- The [tex]\( x^6 \)[/tex] term becomes very large.
- Since the leading coefficient is negative (-5), [tex]\( -5x^6 \)[/tex] becomes very large and negative.
- Thus, as [tex]\( x \)[/tex] approaches positive infinity, the term [tex]\( -5x^6 \)[/tex] dominates the polynomial, causing [tex]\( f(x) \)[/tex] to decrease without bound.
Therefore, as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) decreases.
3. Behavior as [tex]\( x \)[/tex] Approaches Negative Infinity:
When [tex]\( x \)[/tex] becomes a very large negative number:
- Again, the [tex]\( x^6 \)[/tex] term becomes very large because the exponent 6 is even.
- Since the leading coefficient is negative (-5), [tex]\( -5x^6 \)[/tex] still becomes very large and negative because the negative sign in front remains.
- As [tex]\( x \)[/tex] approaches negative infinity, the term [tex]\( -5x^6 \)[/tex] dominates, causing [tex]\( f(x) \)[/tex] to decrease without bound.
Therefore, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) decreases.
### Conclusion:
Based on the analysis above, the correct statements about the polynomial [tex]\( f(x) = -5x^6 + 3x^4 - 2x^3 - 8 \)[/tex] are:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] decreases.
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( y \)[/tex] decreases.
The other statements given in the question are incorrect because the polynomial does not approach zero or increase as [tex]\( x \)[/tex] goes to either positive or negative infinity. The leading term [tex]\( -5x^6 \)[/tex] ensures that [tex]\( f(x) \)[/tex] decreases in both directions towards infinity.
### Step-by-Step Analysis
1. Highest Degree Term Dominance:
The term with the highest degree in the polynomial will dominate the behavior of [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches positive or negative infinity. In this polynomial, the highest degree term is [tex]\( -5x^6 \)[/tex].
2. Behavior as [tex]\( x \)[/tex] Approaches Positive Infinity:
When [tex]\( x \)[/tex] becomes a very large positive number:
- The [tex]\( x^6 \)[/tex] term becomes very large.
- Since the leading coefficient is negative (-5), [tex]\( -5x^6 \)[/tex] becomes very large and negative.
- Thus, as [tex]\( x \)[/tex] approaches positive infinity, the term [tex]\( -5x^6 \)[/tex] dominates the polynomial, causing [tex]\( f(x) \)[/tex] to decrease without bound.
Therefore, as [tex]\( x \)[/tex] approaches positive infinity, [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) decreases.
3. Behavior as [tex]\( x \)[/tex] Approaches Negative Infinity:
When [tex]\( x \)[/tex] becomes a very large negative number:
- Again, the [tex]\( x^6 \)[/tex] term becomes very large because the exponent 6 is even.
- Since the leading coefficient is negative (-5), [tex]\( -5x^6 \)[/tex] still becomes very large and negative because the negative sign in front remains.
- As [tex]\( x \)[/tex] approaches negative infinity, the term [tex]\( -5x^6 \)[/tex] dominates, causing [tex]\( f(x) \)[/tex] to decrease without bound.
Therefore, as [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] (or [tex]\( f(x) \)[/tex]) decreases.
### Conclusion:
Based on the analysis above, the correct statements about the polynomial [tex]\( f(x) = -5x^6 + 3x^4 - 2x^3 - 8 \)[/tex] are:
- As [tex]\( x \)[/tex] approaches negative infinity, [tex]\( y \)[/tex] decreases.
- As [tex]\( x \)[/tex] approaches positive infinity, [tex]\( y \)[/tex] decreases.
The other statements given in the question are incorrect because the polynomial does not approach zero or increase as [tex]\( x \)[/tex] goes to either positive or negative infinity. The leading term [tex]\( -5x^6 \)[/tex] ensures that [tex]\( f(x) \)[/tex] decreases in both directions towards infinity.
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