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Question

$$ 1< $$ For the polynomial function $$ f(x)=-4 x^{4}+8 x^{3} $$, answer the parts a through e. a. Use the Leading Coefficient Test to determine the graph's end behavior. A. The graph of $$ f(x) $$ falls to the left and falls to the right. B. The graph of $$ f(x) $$ rises to the left and rises to the right. C. The graph of $$ f(x) $$ rises to the left and falls to the right. D. The graph of $$ f(x) $$ falls to the left and rises to the right. b. Find the $$ x $$-intercepts. State whether the graph crosses the $$ x $$-axis, or touches the $$ x $$-axis and turns around, at each intercept. The $$ x $$-intercept(s) is/are 0,2 . (Type an integer or a decimal. Use a comma to separate answers as needed. Type each answer only once.) At which $$ x $$-intercept(s) does the graph cross the $$ x $$-axis? Select the correct choice below and, if necessary, fill in the answer box to comi A. The $$ x $$-intercept(s) at which the graph crosses the $$ x $$-axis is/are (Type an integer or a decimal. Use a comma to separate answers as netded. Type each answer only once.) B. There are no $$ x $$-intercepts at which the graph crosses the $$ x $$-axis.

UpStudy AI · Accepted Answer

**a. End Behavior**

The polynomial is  
$$
f(x) = -4x^4 + 8x^3.
$$  
The degree is $$4$$ (even) and the leading coefficient is $$-4$$ (negative). For an even-degree polynomial with a negative leading coefficient, the graph falls on both ends.

Thus, the answer is:  
**A. The graph of $$ f(x) $$ falls to the left and falls to the right.**

---

**b. $$x$$-Intercepts and Behavior at Each**

To find the $$x$$-intercepts, factor the function:

$$
f(x) = -4x^4 + 8x^3 = -4x^3(x-2).
$$

Set $$ f(x) = 0 $$:

$$
-4x^3(x-2) = 0.
$$

This gives us the factors:

1. $$-4x^3 = 0 \quad \Longrightarrow \quad x^3 = 0 \quad \Longrightarrow \quad x = 0,$$ with multiplicity $$3$$.
2. $$x-2 = 0 \quad \Longrightarrow \quad x = 2,$$ with multiplicity $$1$$.

Since both multiplicities (3 and 1) are odd, the graph **crosses** the $$x$$-axis at both intercepts.

Thus, the $$x$$-intercepts are at  
$$
x = 0 \text{ and } x = 2,
$$  
and at both $$x = 0$$ and $$x = 2$$ the graph crosses the $$x$$-axis.

For the selection question, the answer is:  
**A. The $$x$$-intercept(s) at which the graph crosses the $$x$$-axis is/are $$0, 2$$.**
**a. End Behavior** The graph of $$ f(x) = -4x^4 + 8x^3 $$ falls to the left and falls to the right. **b. $$x$$-Intercepts and Behavior** The $$x$$-intercepts are at $$x = 0$$ and $$x = 2$$. At both intercepts, the graph crosses the $$x$$-axis.

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