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Consider the polynomial function

h(x)=x^(6)-3x^(5)-17x^(3)". "
What is the end behavior of the graph of 
h ?
Choose 1 answer:
(A) As 
x rarr oo,h(x)rarr oo, and as 
x rarr-oo, 
h(x)rarr oo.
(B) As 
x rarr oo, 
h(x)rarr-oo, and as 
x rarr-oo,h(x)rarr oo.
(c) As 
x rarr oo, 
h(x)rarr-oo, and as 
x rarr-oo,h(x)rarr-oo.
(D) As 
x rarr oo,h(x)rarr oo, and as 
x rarr-oo, 
h(x)rarr-oo.

Consider the polynomial function\newlineh(x)=x63x517x3 h(x)=x^{6}-3 x^{5}-17 x^{3} \text {. } \newlineWhat is the end behavior of the graph of h h ?\newlineChoose 11 answer:\newline(A) As x,h(x) x \rightarrow \infty, h(x) \rightarrow \infty , and as x x \rightarrow-\infty , h(x) h(x) \rightarrow \infty .\newline(B) As x x \rightarrow \infty , h(x) h(x) \rightarrow-\infty , and as x,h(x) x \rightarrow-\infty, h(x) \rightarrow \infty .\newline(C) As x x \rightarrow \infty , h(x) h(x) \rightarrow-\infty , and as x,h(x) x \rightarrow-\infty, h(x) \rightarrow-\infty .\newline(D) As x,h(x) x \rightarrow \infty, h(x) \rightarrow \infty , and as x x \rightarrow-\infty , h(x) h(x) \rightarrow-\infty .

Full solution

Q. Consider the polynomial function\newlineh(x)=x63x517x3 h(x)=x^{6}-3 x^{5}-17 x^{3} \text {. } \newlineWhat is the end behavior of the graph of h h ?\newlineChoose 11 answer:\newline(A) As x,h(x) x \rightarrow \infty, h(x) \rightarrow \infty , and as x x \rightarrow-\infty , h(x) h(x) \rightarrow \infty .\newline(B) As x x \rightarrow \infty , h(x) h(x) \rightarrow-\infty , and as x,h(x) x \rightarrow-\infty, h(x) \rightarrow \infty .\newline(C) As x x \rightarrow \infty , h(x) h(x) \rightarrow-\infty , and as x,h(x) x \rightarrow-\infty, h(x) \rightarrow-\infty .\newline(D) As x,h(x) x \rightarrow \infty, h(x) \rightarrow \infty , and as x x \rightarrow-\infty , h(x) h(x) \rightarrow-\infty .
  1. Identify Leading Term: Identify the leading term of the polynomial function. The leading term of the polynomial function h(x)=x63x517x3h(x) = x^6 - 3x^5 - 17x^3 is x6x^6, since it has the highest power of xx.
  2. Determine End Behavior: Determine the end behavior based on the leading term.\newlineThe end behavior of a polynomial function is determined by its leading term. Since the leading term is x6x^6, which is an even power, the end behavior will be the same in both directions of the x-axis. As xx approaches positive infinity (xx \rightarrow \infty), x6x^6 will also approach positive infinity. Similarly, as xx approaches negative infinity (xx \rightarrow -\infty), x6x^6 will also approach positive infinity because an even power of a negative number is positive.
  3. Choose Correct Answer: Choose the correct answer based on the end behavior.\newlineBased on the end behavior determined in Step 22, the correct answer is:\newline(A) As xx \rightarrow \infty, h(x)h(x) \rightarrow \infty, and as xx \rightarrow -\infty, h(x)h(x) \rightarrow \infty.

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