Answer :
To determine the end behavior of the polynomial [tex]\( h(x) = -8 x^2 (x+6)^7 (x-9)^4 \)[/tex], we need to analyze how [tex]\( h(x) \)[/tex] behaves as [tex]\( x \)[/tex] approaches positive and negative infinity.
1. Identify the highest degree term:
- The polynomial [tex]\( h(x) \)[/tex] can be broken down into three main parts: [tex]\( x^2 \)[/tex], [tex]\( (x+6)^7 \)[/tex], and [tex]\( (x-9)^4 \)[/tex].
- Each part contributes to the overall degree of the polynomial:
- [tex]\( x^2 \)[/tex] contributes [tex]\( x^2 \)[/tex].
- [tex]\( (x+6)^7 \)[/tex] contributes [tex]\( x^7 \)[/tex]. (For large [tex]\( |x| \)[/tex], the [tex]\( +6 \)[/tex] becomes negligible compared to [tex]\( x \)[/tex]).
- [tex]\( (x-9)^4 \)[/tex] contributes [tex]\( x^4 \)[/tex]. (Again, for large [tex]\( |x| \)[/tex], the [tex]\( -9 \)[/tex] becomes negligible).
2. Combine the highest degree terms:
- Multiplying these together gives the highest degree term:
[tex]\[ h_{\text{leading}}(x) = -8 \cdot x^2 \cdot x^7 \cdot x^4 = -8 x^{2 + 7 + 4} = -8 x^{13} \][/tex]
3. Analyze the leading term for large values of [tex]\( x \)[/tex]:
- The leading term [tex]\( -8 x^{13} \)[/tex] determines the end behavior of the polynomial:
- As [tex]\( x \rightarrow +\infty \)[/tex]:
- Since [tex]\( -8 x^{13} \)[/tex] includes an odd exponent and a negative coefficient, [tex]\( x^{13} \)[/tex] will be very large and positive, and thus [tex]\( -8 x^{13} \)[/tex] will be very large and negative.
- Therefore, [tex]\( h(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex]:
- [tex]\( x^{13} \)[/tex] will be very large and negative because [tex]\( 13 \)[/tex] is an odd power and [tex]\( -8 x^{13} \)[/tex] will be very large and positive.
- Therefore, [tex]\( h(x) \rightarrow +\infty \)[/tex].
In summary:
- As [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( h(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( h(x) \rightarrow +\infty \)[/tex].
To provide specific numerical examples:
- When [tex]\( x = 10 \)[/tex]:
[tex]\[ h(x) \approx -8 \cdot 10^{13} = -80,000,000,000,000 \][/tex]
- When [tex]\( x = -10 \)[/tex]:
[tex]\[ h(x) \approx -8 \cdot (-10)^{13} = 80,000,000,000,000 \][/tex]
Thus, the end behaviors are quantified as:
- [tex]\( h(10) = -80,000,000,000,000 \)[/tex].
- [tex]\( h(-10) = 80,000,000,000,000 \)[/tex].
1. Identify the highest degree term:
- The polynomial [tex]\( h(x) \)[/tex] can be broken down into three main parts: [tex]\( x^2 \)[/tex], [tex]\( (x+6)^7 \)[/tex], and [tex]\( (x-9)^4 \)[/tex].
- Each part contributes to the overall degree of the polynomial:
- [tex]\( x^2 \)[/tex] contributes [tex]\( x^2 \)[/tex].
- [tex]\( (x+6)^7 \)[/tex] contributes [tex]\( x^7 \)[/tex]. (For large [tex]\( |x| \)[/tex], the [tex]\( +6 \)[/tex] becomes negligible compared to [tex]\( x \)[/tex]).
- [tex]\( (x-9)^4 \)[/tex] contributes [tex]\( x^4 \)[/tex]. (Again, for large [tex]\( |x| \)[/tex], the [tex]\( -9 \)[/tex] becomes negligible).
2. Combine the highest degree terms:
- Multiplying these together gives the highest degree term:
[tex]\[ h_{\text{leading}}(x) = -8 \cdot x^2 \cdot x^7 \cdot x^4 = -8 x^{2 + 7 + 4} = -8 x^{13} \][/tex]
3. Analyze the leading term for large values of [tex]\( x \)[/tex]:
- The leading term [tex]\( -8 x^{13} \)[/tex] determines the end behavior of the polynomial:
- As [tex]\( x \rightarrow +\infty \)[/tex]:
- Since [tex]\( -8 x^{13} \)[/tex] includes an odd exponent and a negative coefficient, [tex]\( x^{13} \)[/tex] will be very large and positive, and thus [tex]\( -8 x^{13} \)[/tex] will be very large and negative.
- Therefore, [tex]\( h(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex]:
- [tex]\( x^{13} \)[/tex] will be very large and negative because [tex]\( 13 \)[/tex] is an odd power and [tex]\( -8 x^{13} \)[/tex] will be very large and positive.
- Therefore, [tex]\( h(x) \rightarrow +\infty \)[/tex].
In summary:
- As [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex], [tex]\( h(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \)[/tex] approaches [tex]\( -\infty \)[/tex], [tex]\( h(x) \rightarrow +\infty \)[/tex].
To provide specific numerical examples:
- When [tex]\( x = 10 \)[/tex]:
[tex]\[ h(x) \approx -8 \cdot 10^{13} = -80,000,000,000,000 \][/tex]
- When [tex]\( x = -10 \)[/tex]:
[tex]\[ h(x) \approx -8 \cdot (-10)^{13} = 80,000,000,000,000 \][/tex]
Thus, the end behaviors are quantified as:
- [tex]\( h(10) = -80,000,000,000,000 \)[/tex].
- [tex]\( h(-10) = 80,000,000,000,000 \)[/tex].
