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16
Question 9
1 pts
Use the factored form of the polynomial function below to describe its end behaviors.
\[
x^{2}(x-1)(x+3)(x-2)=0
\]
As $x \rightarrow-\infty, f(x) \rightarrow-\infty$, as $x \rightarrow \infty, f(x) \rightarrow-\infty$
As $x \rightarrow-\infty, f(x) \rightarrow \infty$, as $x \rightarrow \infty, f(x) \rightarrow-\infty$
As $x \rightarrow-\infty, f(x) \rightarrow \infty$, as $x \rightarrow \infty, f(x) \rightarrow \infty$
As $x \rightarrow-\infty, f(x) \rightarrow-\infty$, as $x \rightarrow \infty, f(x) \rightarrow \infty$
- Previous

Step 1 ：The end behavior of a polynomial function can be determined by the degree and the leading coefficient of the polynomial. If the degree of the polynomial is even, and the leading coefficient is positive, the end behavior is: as x approaches negative infinity, f(x) approaches positive infinity, and as x approaches positive infinity, f(x) approaches positive infinity. If the degree is odd and the leading coefficient is positive, the end behavior is: as x approaches negative infinity, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches positive infinity.

Step 2 ：In this case, the polynomial function is \(x^{2}(x-1)(x+3)(x-2)\), which simplifies to \(x^{5} - 5x^{4} + 8x^{3} + 6x^{2}\). The degree of this polynomial is 5 (odd), and the leading coefficient is 1 (positive). Therefore, the end behavior should be: as x approaches negative infinity, f(x) approaches negative infinity, and as x approaches positive infinity, f(x) approaches positive infinity.

Step 3 ：Final Answer: \(\boxed{\text{As } x \rightarrow-\infty, f(x) \rightarrow-\infty, \text{ as } x \rightarrow \infty, f(x) \rightarrow \infty}\)